Fractional Extension of Gould–Hopper–Bell Polynomials Related to Apostol-Type Polynomials and Their Properties

Abstract

This study uses a fractional operator technique to analyze a novel class of special polynomials. These polynomials are designated as fractional Gould–Hopper–Bell–Apostol-type polynomials. We first define the operational expression of the Apostol-type Gould–Hopper–Bell polynomials and then use a suitable fractional operator to generate a new fractional version of these polynomials. The accompanying generating function, series definition, and summation formulas are also derived. Furthermore, certain symmetry identities and monomiality results are investigated. The study also identifies specific members of this fractional family, such as fractional Gould–Hopper–Bell–Apostol–Bernoulli polynomials, fractional Gould–Hopper–Bell–Apostol–Euler polynomials, and fractional Gould–Hopper–Bell–Apostol–Genocchi polynomials, and finds similar results for each. The study makes use of Mathematica to display computational results, zero distributions, and graphical demonstrations for a specific case of the established class.

1. Introduction

Special functions, developed over centuries and supported by a vast literature, are defined with nuance and possess wide applicability in mathematics and other fields. These functions have historically been essential in the advancement of pure and applied mathematics, engineering, physics, and other fields that depend on mathematical approaches as core instruments.

For many years, the study of special numbers and polynomials has been an important part of mathematical analysis, interacting with other disciplines such as mathematical physics, fractional calculus, and differential and integral equations. Furthermore, these special polynomials facilitate the seamless derivation of several valuable identities and play a crucial role in establishing new classifications of special polynomials. Due to their widespread application in a variety of mathematical contexts, the Apostol-type, Gould–Hopper, and Bell polynomials are particularly noteworthy.

This study will constantly utilize the following notations: let $\mathbb{R}$ be the set of real numbers, ${\mathbb{R}}^{+}$ the set of positive real numbers, $\mathbb{C}$ the set of complex numbers, $\mathbb{Z}$ the set of integers, $\mathbb{N}$ the set of positive integers, and ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ the set of non-negative integers.

The Gould–Hopper polynomials (GHP), represented by ${\displaystyle {\mathcal{H}}_{\nu }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2)$ [1], are defined by

$$e^{{\vartheta }_1\omega +{\vartheta }_2{\omega }^r}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathcal{H}}_{\nu }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

For $r=2$, GHP ${\displaystyle {\mathcal{H}}_{\nu }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2)$ reduces to the Hermite Kampé de Fériet polynomials ${\displaystyle {\mathcal{H}}_{\nu }^{\left(2\right)}}({\vartheta }_1,{\vartheta }_2)$ [2], which are given as:

$$e^{{\vartheta }_1\omega +{\vartheta }_2{\omega }^2}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathcal{H}}_{\nu }^{\left(2\right)}}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

The generalized Bell polynomials, represented as $\mathcal{B}e{l}_{\nu }({\vartheta }_1,{\vartheta }_2)$ [3], are defined by

$$e^{{\vartheta }_1\omega +{\vartheta }_2(e^{\omega }-1)}={\displaystyle \sum _{\nu =0}^{\infty }}\mathcal{B}e{l}_{\nu }({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(1)

Taking ${\vartheta }_1=0$ in (1), we get

$$e^{{\vartheta }_2(e^{\omega }-1)}={\displaystyle \sum _{\nu =0}^{\infty }}\mathcal{B}e{l}_{\nu }\left({\vartheta }_2\right){\displaystyle \frac{{\omega }^{\nu }}{\nu !}},$$

(2)

where $\mathcal{B}e{l}_{\nu }\left({\vartheta }_2\right)$ denotes the classical Bell polynomials [4].

The Bernoulli, Euler, and Genocchi polynomials of the Apostol type and order $\epsilon$ [5,6,7] are defined, respectively, by

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\omega }{\zeta e^{\omega }-1}}\right)}^{\epsilon }{e}^{\vartheta \omega }={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathfrak{B}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\left(\right|\omega +log\zeta |<2\pi ,1^{\epsilon }:=1),\end{array}$$

(3)

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{2}{\zeta e^{\omega }+1}}\right)}^{\epsilon }{e}^{\vartheta \omega }={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathcal{E}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\left(\right|\omega +log\zeta |<\pi ,1^{\epsilon }:=1),\end{array}$$

(4)

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{2\omega }{\zeta e^{\omega }+1}}\right)}^{\epsilon }{e}^{\vartheta \omega }={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathcal{G}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\left(\right|\omega +log\zeta |<\pi ,1^{\epsilon }:=1),\end{array}$$

(5)

where $\epsilon$ and $\zeta$ are arbitrary complex parameters. When $\vartheta =0$ in (3), (4) and (5), we get

$${\displaystyle {\mathfrak{B}}_{\nu }^{\left(\epsilon \right)}}(0;\zeta )={\displaystyle {\mathfrak{B}}_{\nu }^{\left(\epsilon \right)}}\left(\zeta \right),{\displaystyle {\mathcal{E}}_{\nu }^{\left(\epsilon \right)}}(0;\zeta )={\displaystyle {\mathcal{E}}_{\nu }^{\left(\epsilon \right)}}\left(\zeta \right)\mathrm{and}{\displaystyle {\mathcal{G}}_{\nu }^{\left(\epsilon \right)}}(0;\zeta )={\displaystyle {\mathcal{G}}_{\nu }^{\left(\epsilon \right)}}\left(\zeta \right),$$

which represent the Bernoulli, Euler, and Genocchi numbers of the Apostol type and order $\epsilon$, respectively.

The class of generalized Apostol-type polynomials (UATP) ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\alpha ,\zeta ,a,b)$ [8] is defined as

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{e}^{\vartheta \omega }={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\left(|\omega +blog\left({\displaystyle \frac{\zeta }{a}}\right)|<2\pi ,\alpha \in {\mathbb{N}}_0,a,b\in \mathbb{R}\setminus \left\{0\right\},\epsilon ,\zeta \in \mathbb{C}\right),\end{array}$$

(6)

where ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(0;\alpha ,\zeta ,a,b):={\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\alpha ,\zeta ,a,b)$ represents the generalized Apostol-type numbers, which are given as:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\end{array}$$

(7)

Also, we note that

$${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;1,\zeta ,1,1)={\displaystyle {\mathfrak{B}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta ),{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;0,\zeta ,-1,1)={\displaystyle {\mathcal{E}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta )$$

and

$${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}\left(\vartheta ;1,{\displaystyle \frac{\zeta }{2}},-{\displaystyle \frac{1}{2}},1\right)={\displaystyle {\mathcal{G}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta ).$$

The Bernoulli, Euler, and Genocchi polynomials of the Apostol type are among the several classical polynomial sequences that fall under the class of generalized Apostol-type polynomials. Considerable families of hybrid special polynomials linked to Apostol-type polynomials have been introduced and investigated in recent studies, for example, Srivastava et al. [9] derived some identities and relation involving the modified degenerate Hermite-based Apostol–Bernoulli and Apostol–Euler polynomials. Luo et al. [5] studied some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials. Srivastava et al. [10] discussed the truncated-exponential based Apostol-type polynomials. Araci et al. [11] established some unified formulas involving generalized-Apostol-type-Gould–Hopper polynomials and multiple power sums. Ramírez and Cesarano [12] applied the monomiality principle to introduce some results on a family of Apostol–Hermite–Bernoulli-type polynomials. Cesarano et al. [13] examined the Apostol-type Hermite degenerated polynomials. These hybrid polynomials are of great importance because they have important mathematical qualities, including differential equations, generating functions, series and determinant definitions, and other fundamental traits.

Recently, Sidaoui et al. [14] introduced Apostol-type Gould–Hopper–Bell polynomials (GHBlATP) ${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)$ of order $\epsilon$, which are defined by

$${\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{e}^{{\vartheta }_1\omega +{\vartheta }_2{\omega }^r+z(e^{\omega }-1)}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}},$$

(8)

$$\left(|\omega +blog\left({\displaystyle \frac{\zeta }{a}}\right)|<2\pi ,\alpha \in {\mathbb{N}}_0,a,b\in \mathbb{R}\setminus \left\{0\right\},\epsilon ,\zeta \in \mathbb{C}\right).$$

The series definitions of the GHBlATP ${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)$ of order $\epsilon$ are given as:

$${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b)\mathcal{B}e{l}_{\kappa }\left(z\right);$$

(9)

$${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathsf{P}}_{\nu -\kappa }^{\left(\epsilon \right)}}{(0;\alpha ,\zeta ,a,b)}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z);$$

(10)

$${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)=\nu !{\displaystyle \sum _{\kappa =0}^{\left[{\displaystyle \frac{\nu }{r}}\right]}}{\displaystyle \frac{{{\displaystyle {\vartheta }_2^{\kappa }}}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu -r\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)}{\kappa !(\nu -r\kappa )!}}.$$

(11)

By substituting $z=0$ in (8), we obtain the unified Gould–Hopper–Apostol-type polynomials ${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b)$ of order $\epsilon$ [15]:

$${\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,0;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{e}^{{\vartheta }_1\omega +{\vartheta }_2{\omega }^r}={\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(12)

Setting ${\vartheta }_2=0$ in (8), we get the unified Bell–Apostol-type polynomials of two variables (2VUBlAP) ${}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)$ of order $\epsilon$ as:

$${\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,0,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{e}^{{\vartheta }_1\omega +z(e^{\omega }-1)}={\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(13)

Setting ${\vartheta }_1={\vartheta }_2=0$ in (8), we get the unified Bell–Apostol-type polynomials ${}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(z;\alpha ,\zeta ,a,b)$ of order $\epsilon$ as:

$${\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}(0,0,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{e}^{z(e^{\omega }-1)}={\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(14)

Substituting $r=2$ into (8), we derive the Hermite Kampé de Fériet–Bell–Apostol-type polynomials ${}_{\mathcal{H}\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{(\epsilon ,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)$ of order $\epsilon$ as:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}& ={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{e}^{{\vartheta }_1\omega +{\vartheta }_2{\omega }^2+z(e^{\omega }-1)}\hfill \\ & ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

(15)

Setting $\epsilon =0$ in (8) yields the Gould–Hopper–Bell polynomials ${}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z)$, which are defined by [16]:

$${\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(0,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}=e^{{\vartheta }_1\omega +{\vartheta }_2{\omega }^r+z(e^{\omega }-1)}={\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(16)

The combination of integral transforms and fractional calculus has developed into an important cross-disciplinary field during the past forty years, primarily due to its extensive application in a variety of technical and scientific fields. Furthermore, fractional calculus provides an effective tool for identifying new characteristics and connections among special polynomials. It is possible to develop novel extensions of hybrid forms of special polynomials by utilizing fractional operators combined with existing special polynomials [17,18,19].

Fractional operators provide a more precise depiction of intricate systems that cannot be represented by integer-order derivatives. Consequently, they possess substantial applications across many domains, encompassing multiple branches of mathematics, physics [20], finance [21], and engineering [22]. For example, the behavior of viscoelastic materials, biological systems, and electrical networks can be characterized using fractional operators [23]. Furthermore, fractional operators are utilized in electromagnetics to characterize the behavior of electromagnetic waves in materials exhibiting fractional-order dielectric and magnetic characteristics [24]. Additional uses of fractional calculus and special polynomials are presented in [25,26,27,28].

The Euler integral, which was introduced by Srivastava and Manocha [29], is one of the helpful fractional operators on special polynomials. Through the use of the Euler integral as a fundamental tool, researchers were able to discover new avenues for the advancement of special polynomial theory. By utilizing the Euler integral formulation, Dattoli et al. [30,31] have significantly advanced the theory and use of special polynomials. The Euler integral [29] (p. 218) is defined as follows:

$${\eta }^{-\sigma }={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma -1}d\xi ,min\left\{\Re \left(\eta \right),\Re \left(\sigma \right)\right\}>0,$$

(17)

which leads to the following [30]:

$${\left(\eta -{\displaystyle \frac{\partial }{\partial \vartheta }}\right)}^{-\sigma }f\left(\vartheta \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma -1}exp\left(\xi {\displaystyle \frac{\partial }{\partial \vartheta }}\right)f\left(\vartheta \right)d\xi$$

and consequently, we can get

$${\left(\eta -{\displaystyle \frac{{\partial }^r}{\partial {\vartheta }^r}}\right)}^{-\sigma }f\left(\vartheta \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma -1}exp\left(\xi {\displaystyle \frac{{\partial }^r}{\partial {\vartheta }^r}}\right)f\left(\vartheta \right)d\xi .$$

The effective utilization of fractional operators can be accomplished by integrating the features of exponential operators with carefully selected integral representations.

This study’s originality lies in the development of a generalized class of special polynomials. This investigation advances the utilization of fractional methods to produce a comprehensive class of polynomials that builds upon and extends existing families (e.g., Gould–Hopper, Gould–Hopper–Bernoulli, Gould–Hopper–Euler, and Gould–Hopper–Genocchi polynomials [1,32,33,34]; Bell, Gould–Hopper–Bell, Bell–Bernoulli, Bell–Euler, and Bell–Genocchi polynomials [3,4,35,36]; Apostol-type, Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi polynomials [5,6,8,37]; Bell–Apostol–Bernoulli, Bell–Apostol–Euler, and Bell–Apostol–Genocchi polynomials [38,39]; Gould-Hopper-Apostol-Bernoulli, Gould–Hopper–Apostol–Euler, and Gould–Hopper–Apostol–Genocchi polynomials [15,16,39]; and Gould–Hopper–Bell–Apostol-type polynomials [14]). This family advances the theory of special functions by presenting and meticulously examining a new, extensive class of polynomials. This approach shows how fractional operational methods can be useful and flexible when building and analyzing generalized polynomial structures. It also shows how associating special polynomials with fractional calculus can be useful in areas where fractional derivatives are useful and important.

This work is motivated by the contributions of Dattoli et al. [30], Dattoli [31], Khan et al. [19,40], Yasmin and Muhyi [18], Wani [41], and Zayed et al. [42], who emphasized the significance of fractional operators in enhancing both theoretical and practical approaches within the domain of special functions. This study uses a fractional operator technique to introduce a fractional family of special polynomials, referred as the fractional Gould–Hopper–Bell–Apostol-type polynomials. In Section 2, we initially employ operational approaches to derive the operational expression of the Gould–Hopper–Bell–Apostol-type polynomials. We further enhance this technique by integrating fractional operators to formulate a more comprehensive generalization of these polynomials. Corresponding generating functions and series expressions are also developed. Section 3 concentrates on developing pertinent summation relations. In Section 4, we derive specific symmetry identities for these fractional polynomials. Section 5 concentrates on deriving the multiplicative and derivative operators, together with the differential equation that the established family satisfies. Section 6 highlights specific members within the introduced family, including the fractional Gould–Hopper–Bell–Apostol–Bernoulli, Euler, and Genocchi polynomials, demonstrating comparable outcomes for each. Section 7 offers visual and numerical insights into these polynomials’ behavior by presenting computational results, zero distributions, and graphical visualizations for a specific class of fractional Gould–Hopper–Bell–Apostol-type polynomials, utilizing Mathematica. Appendix A provides a comprehensive summary of the polynomials and their corresponding symbols.

2. Fractional Gould–Hopper–Bell–Apostol-Type Polynomials

This section begins by establishing an operational relationship between GHBlATP ${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)$ and 2VUBlAP ${}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)$. Subsequently, utilizing the acquired operational representation and considering the Euler integral, we formulate the fractional Gould–Hopper–Bell–Apostol-type polynomials.

Upon differentiating expression (8) r times with respect to ${\vartheta }_1$, we obtain

$${\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\left\{{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)\right\}=\left\{\begin{array}{cc}{\displaystyle \frac{\nu !{\displaystyle {\mathbb{P}}_{\nu -r}^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)}{(\nu -r)!}},\hfill & \nu \ge r;\hfill \\ 0,\hfill & 0\le \nu <r.\hfill \end{array}\right.$$

(18)

Next, on differentiating expression (8) with respect to ${\vartheta }_2$, we find

$${\displaystyle \frac{\partial }{\partial {\vartheta }_2}}\left\{{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)\right\}=\left\{\begin{array}{cc}{\displaystyle \frac{\nu !{\displaystyle {\mathbb{P}}_{\nu -r}^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)}{(\nu -r)!}},\hfill & \nu \ge r;\hfill \\ 0,\hfill & 0\le \nu <r.\hfill \end{array}\right.$$

(19)

From (18) and (19), we obtain

$${\displaystyle \frac{\partial }{\partial {\vartheta }_2}}\left\{{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)\right\}={\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\left\{{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)\right\}.$$

(20)

From (13), it follows that

$${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,0,z;\alpha ,\zeta ,a,b)={}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b).$$

(21)

Solving (20) subject to initial condition (21), we get

$${\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b)=exp\left({\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)\left\{{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)\right\}.$$

(22)

We now derive the fractional Gould–Hopper–Bell–Apostol-type polynomials by demonstrating the subsequent results:

Theorem 1.

The fractional Gould–Hopper–Bell–Apostol-type polynomials, denoted as ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ),$ are defined by

$${\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)}^{-\sigma }\left\{{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)\right\}={\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ).$$

(23)

Proof. 

Substituting $\eta$ with $\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)$ in (17) and subsequently applying the derived equation to ${}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)$ yields the following transformation:

$$\begin{array}{cc}\hfill {\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)}^{-\sigma }& \left\{{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)\right\}\hfill \\ & ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma -1}exp\left(\xi {\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)\left\{{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)\right\}d\xi ,\hfill \end{array}$$

which, after applying the operational relation (22), results in

$$\begin{array}{cc}\hfill {\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)}^{-\sigma }& \left\{{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)\right\}\hfill \\ & ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma -1}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,\xi {\vartheta }_2,z;\alpha ,\zeta ,a,b)d\xi .\hfill \end{array}$$

(24)

In (24), the integral transform on the right side generates a new family of special polynomials, termed the fractional Gould–Hopper–Bell–Apostol-type polynomials (FGHBlATP). Let these polynomials be represented as ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$, yielding

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma -1}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,\xi {\vartheta }_2,z;\alpha ,\zeta ,a,b)d\xi .$$

(25)

From (24) and (25), we get assertion (23). □

Remark 1.

For $z=0$, FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ reduce to the fractional Gould–Hopper–Apostol-type polynomials (FGHATP) ${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )$, which satisfy the operational representation:

$${\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)}^{-\sigma }{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1;\alpha ,\zeta ,a,b)={}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ).$$

Remark 2.

For $r=2$, FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ reduce to the fractional Hermite Kampé de Fériet–Bell–Apostol-type polynomials (FHKDFBlATP) ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$, which satisfy the operational representation:

$${\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^2}{\partial {\displaystyle {\vartheta }_1^2}}}\right)}^{-\sigma }\left\{{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)\right\}={\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ).$$

Next, the generating function for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ is established by proving the following theorem:

Theorem 2.

The following generating function for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ holds true:

$${\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(26)

Proof. 

By summing both sides of (25) after multiplying by ${\displaystyle \frac{{\omega }^{\nu }}{\nu !}}$, we derive

$${\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma -1}\left({\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu }^{(\epsilon ,r)}}({\vartheta }_1,\xi {\vartheta }_2,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)d\xi .$$

(27)

Applying (8) to the right-hand side of (27), we have

$${\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{e}^{{\vartheta }_1\omega +z(e^{\omega }-1)}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle {\int }_0^{\infty }}{e}^{-(\eta -{\vartheta }_2{\omega }^r)\xi }{\xi }^{\sigma -1}d\xi .$$

(28)

Finally, utilizing identity (17) on the right-hand side of (28) yields assertion (26). □

Remark 3.

Setting $z=0$ in (26), we get the generating function of FGHATP ${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )$:

$${\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega }}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}={\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(29)

Remark 4.

Setting $r=2,\epsilon =1$ in (26), we get the generating function of FHKDFBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(1,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$:

$$\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right){\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^2)}^{\sigma }}}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(1,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

Remark 5.

Setting $\epsilon =0$ in (26), we get the generating function of the fractional Gould–Hopper–Bell polynomials (FGHBlP) ${}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )$:

$${\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}={\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(30)

The following results are established to present the series representations for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$:

Theorem 3.

For FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$, the following series representation holds true:

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \frac{\nu !}{{\eta }^{\sigma }}}{\displaystyle \sum _{\kappa =0}^{\left[{\displaystyle \frac{\nu }{r}}\right]}}{\displaystyle \frac{{\left(\sigma \right)}_{\kappa }{\displaystyle {\vartheta }_2^{\kappa }}{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu -r\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)}{{\eta }^{\kappa }\kappa !(\nu -r\kappa )!}}.$$

(31)

Proof. 

In view of (11), (25) becomes

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \frac{\nu !}{\mathrm{\Gamma }\left(\sigma \right)}}{\displaystyle \sum _{\kappa =0}^{\left[{\displaystyle \frac{\nu }{r}}\right]}}{\displaystyle \frac{{\displaystyle {\vartheta }_2^{\kappa }}{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu -r\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b)}{\kappa !(\nu -r\kappa )!}}{\displaystyle {\int }_0^{\infty }}{e}^{-\eta \xi }{\xi }^{\sigma +\kappa -1}d\xi .$$

(32)

Applying (17) to the right-hand side of (32) yields (31). □

Remark 6.

Setting $z=0$ in (31), we get the series representation that is satisfied by FGHATP ${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )$:

$${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )={\displaystyle \frac{\nu !}{{\eta }^{\sigma }}}{\displaystyle \sum _{\kappa =0}^{\left[{\displaystyle \frac{\nu }{r}}\right]}}{\displaystyle \frac{{\left(\sigma \right)}_{\kappa }{\displaystyle {\vartheta }_2^{\kappa }}{\displaystyle {\mathsf{P}}_{\nu -r\kappa }^{\left(\epsilon \right)}}({\vartheta }_1;\alpha ,\zeta ,a,b)}{{\eta }^{\kappa }\kappa !(\nu -r\kappa )!}}.$$

Remark 7.

Setting $r=2,\epsilon =1$ in (31), we get the series representation for FHKDFBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(1,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$:

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(1,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \frac{\nu !}{{\eta }^{\sigma }}}{\displaystyle \sum _{\kappa =0}^{\left[{\displaystyle \frac{\nu }{2}}\right]}}{\displaystyle \frac{{\left(\sigma \right)}_{\kappa }{\displaystyle {\vartheta }_2^{\kappa }}{}_{\mathcal{B}el}{\mathsf{P}}_{\nu -2\kappa }({\vartheta }_1,z;\alpha ,\zeta ,a,b)}{{\eta }^{\kappa }\kappa !(\nu -2\kappa )!}}.$$

Remark 8.

Setting $\epsilon =0$ in (31), we get the series representation for FGHBlP ${}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )$:

$${}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )={\displaystyle \frac{\nu !}{{\eta }^{\sigma }}}{\displaystyle \sum _{\kappa =0}^{\left[{\displaystyle \frac{\nu }{2}}\right]}}{\displaystyle \frac{{\left(\sigma \right)}_{\kappa }\mathcal{B}e{l}_{\nu -2\kappa }({\vartheta }_1,z)}{{\eta }^{\kappa }\kappa !(\nu -2\kappa )!}}.$$

Theorem 4.

FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ satisfy the following series expression:

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )\mathcal{B}e{l}_{\kappa }\left(z\right).$$

(33)

Proof. 

Utilizing (2), (26), and (29), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}& ={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}\hfill \\ & =\left({\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left({\displaystyle \sum _{\nu =0}^{\infty }}\mathcal{B}e{l}_{\nu }\left(z\right){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\hfill \\ & ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )\mathcal{B}e{l}_{\kappa }\left(z\right){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

Equating coefficients of similar powers of $\omega$ yields the result presented in (33). □

Theorem 5.

The following explicit formula for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ holds true:

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathsf{P}}_{\nu -\kappa }^{\left(\epsilon \right)}}(\alpha ,\zeta ,a,b){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta ).$$

(34)

Proof. 

Equations (7), (26), and (30), together with a similar argument to that employed in the proof of Theorem 4, lead to assertion (34). □

3. Summation Formulae

This section presents valuable identities for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$, emphasizing the summation formulae established by the subsequent results.

Theorem 6.

The FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ satisfy the following summation identity:

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z+\rho ;\alpha ,\zeta ,a,b,\eta )={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu -\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}(h,{\vartheta }_2,\rho ;\eta ).$$

(35)

Proof. 

Replacing ${\vartheta }_1,z,\sigma$ with ${\vartheta }_1+h,z+\rho ,\sigma +\varsigma$, respectively, in (26) and employing (13) and (30), we get

$$\begin{array}{c}\sum _{\nu =0}^{\infty }{\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}({\vartheta }_1+h,{\vartheta }_2,z+\rho ;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{\displaystyle \frac{e^{({\vartheta }_1+h)\omega +(z+\rho )(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}\hfill \\ =\left({\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left({\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}(h,{\vartheta }_2,\rho ;\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\hfill \\ ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu -\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\alpha ,\zeta ,a,b){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}(h,{\vartheta }_2,\rho ;\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

(36)

From (36), we get the asserted result (35). □

Theorem 7.

For $\nu ,\alpha \in {\mathbb{N}}_0$ and $\epsilon ,\iota ,\zeta \in \mathbb{C}$, we have

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon +\iota ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\kappa }^{\left(\iota \right)}}(z;\alpha ,\zeta ,a,b).$$

(37)

Proof. 

Utilizing (14), (26) and (29), we have

$$\begin{array}{c}{\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon +\iota ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon +\iota }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}\hfill \\ =\left({\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left({\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\nu }^{\left(\iota \right)}}(z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\hfill \\ ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{B}el}{\displaystyle {\mathsf{P}}_{\kappa }^{\left(\iota \right)}}(z;\alpha ,\zeta ,a,b){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

(38)

From (38), we get claimed result (37). □

Theorem 8.

FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ satisfy the following summation identity:

$${\displaystyle {\mathbb{P}}_{\nu ,\sigma +\upsilon }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\upsilon }^{\left(r\right)}}(h,{\vartheta }_2,z;\eta ).$$

(39)

Proof. 

Utilizing (26), (29) and (30), we have

$$\begin{array}{c}{\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma +\upsilon }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{\displaystyle \frac{e^{({\vartheta }_1+h)\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma +\upsilon }}}\hfill \\ =\left({\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left({\displaystyle \sum _{\nu =0}^{\infty }}{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\upsilon }^{\left(r\right)}}(h,{\vartheta }_2,z;\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\hfill \\ ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\upsilon }^{\left(r\right)}}(h,{\vartheta }_2,z;\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

(40)

From (40), we arrive at asserted result (39). □

The second-kind generalized Stirling numbers are defined as follows [9,37]:

$${\displaystyle \sum _{\nu =0}^{\infty }}S(\nu ,\epsilon ,a,b,\zeta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}={\displaystyle \frac{{\left({\zeta }^b{e}^{\omega }-a^b\right)}^{\epsilon }}{\epsilon !}}.$$

(41)

Theorem 9.

The following explicit formula for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ holds true:

$${\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )S(\kappa ,\epsilon ,a,b,\zeta )={\displaystyle \frac{\nu !2^{\epsilon (1-\alpha )}{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu -\alpha \epsilon ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )}{\epsilon !(\nu -\alpha \epsilon )!}}.$$

(42)

Proof. 

From (26), we have

$${\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}={\left({\displaystyle \frac{{\zeta }^b{e}^{\omega }-a^b}{2^{1-\alpha }{\omega }^{\alpha }}}\right)}^{\epsilon }{\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(43)

Applying (30) and (41) to (43), we get

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\nu =0}^{\infty }}& {}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\hfill \\ & ={\displaystyle \frac{\epsilon !}{{\left(2^{1-\alpha }{\omega }^{\alpha }\right)}^{\epsilon }}}\left({\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left({\displaystyle \sum _{\nu =0}^{\infty }}S(\nu ,\epsilon ,a,b,\zeta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\hfill \\ & ={\displaystyle \frac{\epsilon !}{{\left(2^{1-\alpha }{\omega }^{\alpha }\right)}^{\epsilon }}}{\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )S(\kappa ,\epsilon ,a,b,\zeta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

(44)

From (44), we arrive at asserted result (42). □

Theorem 10.

For FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$, we have

$$\begin{array}{cc}\hfill & {\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\hfill \\ & ={\displaystyle \frac{\zeta }{2}}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\mathcal{E}}_{\nu -\kappa }\left(\zeta \right)\left\{{\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )+{\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\right\}.\hfill \end{array}$$

(45)

Proof. 

From (26), we have

$$\begin{array}{cc}\hfill (\zeta e^{\omega }& +1){\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\hfill \\ & =\zeta {\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}+{\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}},\hfill \end{array}$$

(46)

which can be written as:

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\hfill \\ & ={\displaystyle \frac{\zeta }{2}}\left({\displaystyle \sum _{\nu =0}^{\infty }}{\mathcal{E}}_{\nu }\left(\zeta \right){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left({\displaystyle \sum _{\nu =0}^{\infty }}\left({\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )+{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\right){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right).\hfill \end{array}$$

(47)

From (47), we arrive at asserted result (45). □

4. Symmetry Identities

In this section, we present specific symmetric identities for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$.

Theorem 11.

For $\nu ,\alpha \in {\mathbb{N}}_0,p,q\in \mathbb{N},a,b\in \mathbb{R}\setminus \left\{0\right\}$ and $\epsilon ,\zeta \in \mathbb{C}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)& p^{\nu -\kappa }{q}^{\kappa }{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ).\hfill \end{array}$$

(48)

Proof. 

Let us consider

$$\begin{array}{cc}\hfill \psi \left(\omega \right)& ={\left({\displaystyle \frac{2^{2(1-\alpha )}{\left(pq{\omega }^2\right)}^{\alpha }}{({\zeta }^b{e}^{p\omega }-a^b)({\zeta }^b{e}^{q\omega }-a^b)}}\right)}^{\epsilon }{\displaystyle \frac{e^{2pq{\vartheta }_1\omega +z(e^{p\omega }-1)+z(e^{q\omega }-1)}}{{\left((\eta -{\vartheta }_2{p}^r{\omega }^r)(\eta -{\vartheta }_2{q}^r{\omega }^r)\right)}^{\sigma }}}\hfill \\ & =\left({\displaystyle \sum _{\nu =0}^{\infty }}{p}^{\nu }{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left({\displaystyle \sum _{\nu =0}^{\infty }}{q}^{\nu }{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\hfill \\ & ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)p^{\nu -\kappa }{q}^{\kappa }{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

(49)

Similarly, we can get

$$\begin{array}{c}\hfill \psi \left(\omega \right)={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\end{array}$$

(50)

From (49) and (50), we get the asserted result (48). □

Remark 9.

Setting $z=0$ in (48) yields the following formula for FGHATP ${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)& p^{\nu -\kappa }{q}^{\kappa }{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ).\hfill \end{array}$$

(51)

Remark 10.

Setting $r=2,\epsilon =1$ in (48), we get the following formula for FHKDFBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(1,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)& p^{\nu -\kappa }{q}^{\kappa }{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(1,2)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(1,2)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(1,2)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(1,2)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ).\hfill \end{array}$$

(52)

Remark 11.

Setting $\epsilon =0$ in (48), we get the following result for FGHBlP ${}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)& p^{\nu -\kappa }{q}^{\kappa }{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu -\kappa ,\sigma }^{\left(r\right)}}(q{\vartheta }_1,{\vartheta }_2,z;\eta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}(p{\vartheta }_1,{\vartheta }_2,z;\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu -\kappa ,\sigma }^{\left(r\right)}}(p{\vartheta }_1,{\vartheta }_2,z;\eta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}(q{\vartheta }_1,{\vartheta }_2,z;\eta ).\hfill \end{array}$$

(53)

Theorem 12.

For $\nu ,\alpha \in {\mathbb{N}}_0,p,q\in \mathbb{N},a,b\in \mathbb{R}\setminus \left\{0\right\}$ and $\epsilon ,\zeta \in \mathbb{C}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}& {\displaystyle \sum _{\varpi =0}^{q-1}}{\displaystyle \sum _{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{a}^{b(p+q-2)}{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}{\displaystyle \sum _{\varpi =0}^{p-1}}{\displaystyle \sum _{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{a}^{b(p+q-2)}{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ).\hfill \end{array}$$

(54)

Proof. 

Consider

$$\begin{array}{cc}\hfill \psi \left(\omega \right)& ={\left({\displaystyle \frac{2^{2(1-\alpha )}{\left(pq{\omega }^2\right)}^{\alpha }}{({\zeta }^b{e}^{p\omega }-a^b)({\zeta }^b{e}^{q\omega }-a^b)}}\right)}^{\epsilon }{\displaystyle \frac{({\zeta }^{bq}{e}^{pq\omega }-a^{bq})({\zeta }^{bp}{e}^{pq\omega }-a^{bp})e^{2pq{\vartheta }_1\omega +z(e^{p\omega }-1)+z(e^{q\omega }-1)}}{{\left((\eta -{\vartheta }_2{p}^r{\omega }^r)(\eta -{\vartheta }_2{q}^r{\omega }^r)\right)}^{\sigma }({\zeta }^b{e}^{p\omega }-a^b)({\zeta }^b{e}^{q\omega }-a^b)}}\hfill \\ & =\left({\displaystyle \sum _{\nu =0}^{\infty }}{p}^{\nu }{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right)\left(a^{b(q-1)}{\displaystyle \sum _{\varpi =0}^{q-1}}{\left({\displaystyle \frac{\zeta }{a}}\right)}^{b\varpi }{e}^{p\varpi \omega }\right)\hfill \\ & \times \left({\displaystyle \sum _{\kappa =0}^{\infty }}{q}^{\kappa }{\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\kappa }}{\kappa !}}\right)\left(a^{b(p-1)}{\displaystyle \sum _{\varsigma =0}^{p-1}}{\left({\displaystyle \frac{\zeta }{a}}\right)}^{b\varsigma }{e}^{q\varsigma \omega }\right)\hfill \\ & ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}{\displaystyle \sum _{\varpi =0}^{q-1}}{\displaystyle \sum _{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{a}^{b(p+q-2)}{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

(55)

Similarly, we can get

$$\begin{array}{c}\hfill \psi \left(\omega \right)={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle \sum _{\kappa =0}^{\nu }}{\displaystyle \sum _{\varpi =0}^{p-1}}{\displaystyle \sum _{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{a}^{b(p+q-2)}{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\end{array}$$

(56)

From (55) and (56), we get the asserted result (54). □

Remark 12.

Setting $z=0$ in (54) yields the following formula for FGHATP ${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}& {\displaystyle \sum _{\varpi =0}^{q-1}}{\displaystyle \sum _{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{a}^{b(p+q-2)}{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}{\displaystyle \sum _{\varpi =0}^{p-1}}{\displaystyle \sum _{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{a}^{b(p+q-2)}{}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta ).\hfill \end{array}$$

(57)

Remark 13.

Setting $r=2,\epsilon =1$ in (54), we get the following formula for FHKDFBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(1,2)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}& {\displaystyle \sum _{\varpi =0}^{q-1}}{\displaystyle \sum _{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{a}^{b(p+q-2)}{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(1,2)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(1,2)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}{\displaystyle \sum _{\varpi =0}^{p-1}}{\displaystyle \sum _{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{a}^{b(p+q-2)}{\displaystyle {\mathbb{P}}_{\nu -\kappa ,\sigma }^{(1,2)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle {\mathbb{P}}_{\kappa ,\sigma }^{(1,2)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ).\hfill \end{array}$$

(58)

Remark 14.

Setting $\epsilon =0$ in (54), we get the following result for FGHBlP ${}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\kappa =0}^{\nu }}& {\displaystyle \sum _{\varpi =0}^{q-1}}{\displaystyle \sum _{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{a}^{b(p+q-2)}{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu -\kappa ,\sigma }^{\left(r\right)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2,z;\eta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\eta )\hfill \\ & ={\displaystyle \sum _{\kappa =0}^{\nu }}{\displaystyle \sum _{\varpi =0}^{p-1}}{\displaystyle \sum _{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\left({\displaystyle \frac{\zeta }{a}}\right)}^{b(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{a}^{b(p+q-2)}{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu -\kappa ,\sigma }^{\left(r\right)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2,z;\eta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\eta ).\hfill \end{array}$$

(59)

5. Monomiality Principle

The notion of monomiality derives from the idea of poweroid as presented by Steffensen [43]. This concept is reexamined and systematically implemented by Dattoli [44]. Based on the monomiality principle [43,44], a polynomial set ${\beta }_{\nu }\left(\vartheta \right)$$(\nu \in \mathbb{N},\vartheta \in \mathbb{C})$ is classified as quasi-monomial if one can establish “multiplicative” ($\widehat{M}$) and “derivative” ($\widehat{P}$) operators such that

$$\widehat{M}\left\{{\beta }_{\nu }\left(\vartheta \right)\right\}={\beta }_{\nu +1}\left(\vartheta \right),$$

(60)

$$\widehat{P}\left\{{\beta }_{\nu }\left(\vartheta \right)\right\}=\nu {\beta }_{\nu -1}\left(\vartheta \right),$$

(61)

for all $\nu \in \mathbb{N}$. Additionally, these operators satisfy the identity

$$[\widehat{P},\widehat{M}]=\widehat{P}\widehat{M}-\widehat{M}\widehat{P}=\widehat{1}$$

(62)

and therefore reveal the Weyl group structure. If ${\left\{{\beta }_{\nu }\left(\vartheta \right)\right\}}_{\nu \in \mathbb{N}}$ is quasi-monomial, its properties may be obtained using the operators $\widehat{M}$ and $\widehat{P}$. Thus, we have:

$$\widehat{M}\widehat{P}\left\{{\beta }_{\nu }\left(\vartheta \right)\right\}=\nu {\beta }_{\nu }\left(\vartheta \right),$$

(63)

$${\beta }_{\nu }\left(\vartheta \right)={\widehat{M}}^{\nu }\left\{{\beta }_0\left(\vartheta \right)\right\}={\widehat{M}}^{\nu }\left\{1\right\},{\beta }_0\left(\vartheta \right)=1,$$

(64)

$$exp\left(\omega \widehat{M}\right)\left\{1\right\}={\displaystyle \sum _{\nu =0}^{\infty }}{\beta }_{\nu }\left(\vartheta \right){\displaystyle \frac{{\omega }^{\nu }}{\nu !}},\left|\omega \right|<\infty .$$

(65)

Here, we establish the operators for multiplication and differentiation for FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$.

Theorem 13.

The multiplicative and derivative operators associated with FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$, which exhibit their quasi-monomial characteristics, are as follows:

$${\widehat{M}}_{\mathbb{P}}={\vartheta }_1+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^{r-1}}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}+{\displaystyle \frac{\alpha \epsilon }{D_{{\vartheta }_1}}}-{\displaystyle \frac{\epsilon {\zeta }^b{e}^{D_{{\vartheta }_1}}}{{\zeta }^b{e}^{D_{{\vartheta }_1}}-a^b}}$$

(66)

and

$${\widehat{P}}_{\mathbb{P}}=D_{\vartheta 1},$$

(67)

respectively.

Proof. 

Note that

$$D_{{\vartheta }_1}\left\{{\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}\right\}=\omega \left\{{\left({\displaystyle \frac{2^{1-\alpha }{\omega }^{\alpha }}{{\zeta }^b{e}^{\omega }-a^b}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}\right\}.$$

(68)

Differentiating (26) partially with respect to $\omega$ gives

$$\begin{array}{cc}\hfill \left\{{\vartheta }_1+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\omega }^{r-1}}{\eta -{\vartheta }_2{\omega }^r}}+z{e}^{\omega }+{\displaystyle \frac{\alpha \epsilon }{\omega }}-{\displaystyle \frac{\epsilon {\zeta }^b{e}^{\omega }}{{\zeta }^b{e}^{\omega }-a^b}}\right\}& \left\{{\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}\right\}\hfill \\ & ={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{P}}_{\nu +1,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.\hfill \end{array}$$

By employing the identity (68) and comparing the coefficients of each power of $\omega$, the following is obtained:

$$\begin{array}{cc}\hfill \left\{{\vartheta }_1+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^{r-1}}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}+{\displaystyle \frac{\alpha \epsilon }{D_{{\vartheta }_1}}}-{\displaystyle \frac{\epsilon {\zeta }^b{e}^{D_{{\vartheta }_1}}}{{\zeta }^b{e}^{D_{{\vartheta }_1}}-a^b}}\right\}& \left\{{\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )\right\}\hfill \\ & ={\displaystyle {\mathbb{P}}_{\nu +1,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta ).\hfill \end{array}$$

(69)

Using (60) and (61) (for ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$) in (69) and (68), respectively, provide asserted results (66) and (67). □

Theorem 14.

FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ satisfy the following differential equation:

$$\left(\alpha \epsilon +{\vartheta }_1{D}_{{\vartheta }_1}+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^r}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}{D}_{{\vartheta }_1}-{\displaystyle \frac{\epsilon {\zeta }^b{e}^{D_{{\vartheta }_1}}}{{\zeta }^b{e}^{D_{{\vartheta }_1}}-a^b}}{D}_{{\vartheta }_1}-\nu \right){\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )=0.$$

(70)

Proof. 

In view of relation (63) (for ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$), utilizing operators (66) and (67) yields the asserted result (70). □

Remark 15.

Setting $z=0$ in (66), (67), and (70) yields the following multiplicative and derivative operators, and the differential equation for FGHATP ${}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )$:

$${\widehat{M}}_{HP}={\vartheta }_1+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^{r-1}}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+{\displaystyle \frac{\alpha \epsilon }{D_{{\vartheta }_1}}}-{\displaystyle \frac{\epsilon {\zeta }^b{e}^{D_{{\vartheta }_1}}}{{\zeta }^b{e}^{D_{{\vartheta }_1}}-a^b}},$$

(71)

$${\widehat{P}}_{HP}=D_{\vartheta 1}$$

(72)

and

$$\left(\alpha \epsilon +{\vartheta }_1{D}_{{\vartheta }_1}+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^r}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}-{\displaystyle \frac{\epsilon {\zeta }^b{e}^{D_{{\vartheta }_1}}}{{\zeta }^b{e}^{D_{{\vartheta }_1}}-a^b}}{D}_{{\vartheta }_1}-\nu \right){}_{\mathcal{H}}{\displaystyle {\mathsf{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\alpha ,\zeta ,a,b,\eta )=0,$$

(73)

respectively.

6. Special Members of FGHBlATP ${\displaystyle {\mathbb{P}}_{\mathit{\nu },\mathit{\sigma }}^{(\mathit{\epsilon },\mathit{r})}}({\mathit{\vartheta }}_{\mathbf{1}},{\mathit{\vartheta }}_{\mathbf{2}},\mathit{z};\mathit{\alpha },\mathit{\zeta },\mathit{a},\mathit{b},\mathit{\eta })$

This section presents some significant members of FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ such as fractional Gould–Hopper–Bell–Apostol–Bernoulli polynomials (FGHBlABP) ${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$, fractional Gould–Hopper–Bell–Apostol–Euler polynomials (FGHBlAEP) ${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ and fractional Gould–Hopper–Bell–Apostol–Genocchi polynomials (FGHBlAGP) ${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$.

6.1. Fractional Gould–Hopper–Bell–Apostol–Bernoulli Polynomials

For $\alpha =a=b=1$, UATP ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\alpha ,\zeta ,a,b)$ reduce to the Bernoulli polynomials ${\displaystyle {\mathfrak{B}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta )$, i.e., ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;1,\zeta ,1,1)={\displaystyle {\mathfrak{B}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta )$. Consequently, for the same selection of $\alpha ,a$ and b FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ reduce to FGHBlABP ${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ which are defined by:

$${\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)}^{-\sigma }\left\{{}_{\mathcal{B}el}{\displaystyle {\mathfrak{B}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta )\right\}={\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ).$$

(74)

The generating function for FGHBlABP ${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ is given as:

$${\left({\displaystyle \frac{\omega }{\zeta e^{\omega }-1}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(75)

Specific corresponding results for FGHBlABP ${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ are given in

Table 1. Results for FGHBlABP ${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$.

Series	${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )=\frac{\nu !}{{\eta }^{\sigma }}{\displaystyle {\sum }_{\kappa =0}^{\left[\frac{\nu }{r}\right]}}\frac{{\left(\sigma \right)}_{\kappa }{\displaystyle {\vartheta }_2^{\kappa }}{}_{\mathcal{B}el}{\displaystyle {\mathfrak{B}}_{\nu -rk}^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta )}{{\eta }^{\kappa }\kappa !(\nu -r\kappa )!}$
representations	${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathfrak{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta )\mathcal{B}e{l}_{\kappa }\left(z\right)$
	${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathfrak{B}}_{\nu -\kappa }^{\left(\epsilon \right)}}\left(\zeta \right){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )$
Summation	${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z+\rho ;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{B}el}{\displaystyle {\mathfrak{B}}_{\nu -\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}(h,{\vartheta }_2,\rho ;\eta )$
formulas	${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon +\iota ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathfrak{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta ){}_{\mathcal{B}el}{\displaystyle {\mathfrak{B}}_{\kappa }^{(\iota ,r)}}(z;\zeta )$
	${\displaystyle {\mathbb{B}}_{\nu ,\sigma +\upsilon }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathfrak{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta ){}_{\mathcal{H}}{\displaystyle {\mathfrak{B}}_{\kappa ,\upsilon }^{\left(r\right)}}(h,{\vartheta }_2,z;\eta )$
	${\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathbb{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )S(\kappa ,\epsilon ,\zeta )={\displaystyle \frac{\nu !{}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\nu -\epsilon ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )}{\epsilon !(\nu -\epsilon )!}}$
	${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle \frac{\zeta }{2}}{\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\mathcal{E}}_{\nu -\kappa }\left(\zeta \right)\left\{{\displaystyle {\mathbb{B}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+1,{\vartheta }_2,z;\zeta ,\eta )+{\displaystyle {\mathbb{B}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )\right\}$
Symmetry	${\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)p^{\nu -\kappa }{q}^{\kappa }{\displaystyle {\mathbb{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{B}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{\displaystyle {\mathbb{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{B}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
formulas	${\displaystyle {\sum }_{\kappa =0}^{\nu }}{\displaystyle {\sum }_{\varpi =0}^{q-1}}{\displaystyle {\sum }_{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\zeta }^{(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{\displaystyle {\mathbb{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2,z;\zeta \eta ){\displaystyle {\mathbb{B}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle {\sum }_{\kappa =0}^{\nu }}{\displaystyle {\sum }_{\varpi =0}^{p-1}}{\displaystyle {\sum }_{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\zeta }^{(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{\displaystyle {\mathbb{B}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{B}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\zeta ,\eta )$
Multiplicative and	${\widehat{M}}_{\mathbb{B}}={\vartheta }_1+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^{r-1}}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}+{\displaystyle \frac{\epsilon }{D_{{\vartheta }_1}}}-{\displaystyle \frac{\epsilon \zeta e^{D_{{\vartheta }_1}}}{\zeta e^{D_{{\vartheta }_1}}-1}}$
derivative operators	${\widehat{P}}_{\mathbb{B}}=D_{\vartheta 1}$
Differential equation	$\left(\epsilon +{\vartheta }_1{D}_{{\vartheta }_1}+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^r}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}{D}_{{\vartheta }_1}-{\displaystyle \frac{\epsilon \zeta e^{D_{{\vartheta }_1}}}{\zeta e^{D_{{\vartheta }_1}}-1}}{D}_{{\vartheta }_1}-\nu \right){\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )=0$

.

The first seven FGHBlABP ${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ for $\zeta =\epsilon =1,r=\sigma =2$ are listed in

Table 2. The first seven FGHBlABP ${\displaystyle {\mathbb{B}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$.

$\mathit{\nu }$	${\displaystyle {\mathbb{B}}_{\mathit{\nu },\mathit{\sigma }}^{(\mathit{\epsilon },\mathit{r})}}({\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2,\mathit{z};\mathit{\zeta },\mathit{\eta })$
0	${\displaystyle \frac{1}{{\eta }^2}}$
1	$-{\displaystyle \frac{1}{2{\eta }^2}}+{\displaystyle \frac{{\vartheta }_1}{{\eta }^2}}+{\displaystyle \frac{z}{{\eta }^2}}$
2	${\displaystyle \frac{1}{6{\eta }^2}}+{\displaystyle \frac{{\displaystyle {\vartheta }_1^2}}{{\eta }^2}}-{\displaystyle \frac{{\vartheta }_1}{{\eta }^2}}+{\displaystyle \frac{2{\vartheta }_{1}z}{{\eta }^2}}+{\displaystyle \frac{4{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{z^2}{{\eta }^2}}$
3	${\displaystyle \frac{{\displaystyle {\vartheta }_1^3}}{{\eta }^2}}-{\displaystyle \frac{3{\displaystyle {\vartheta }_1^2}}{2{\eta }^2}}+{\displaystyle \frac{3{\displaystyle {\vartheta }_1^2}z}{{\eta }^2}}+{\displaystyle \frac{{\vartheta }_1}{2{\eta }^2}}+{\displaystyle \frac{12{\vartheta }_1{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{3{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{6{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{12{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{z^3}{{\eta }^2}}+{\displaystyle \frac{3z^2}{2{\eta }^2}}$
4	$\begin{array}{c}-{\displaystyle \frac{1}{30{\eta }^2}}+{\displaystyle \frac{{\displaystyle {\vartheta }_1^4}}{{\eta }^2}}-{\displaystyle \frac{2{\displaystyle {\vartheta }_1^3}}{{\eta }^2}}+{\displaystyle \frac{4{\displaystyle {\vartheta }_1^3}z}{{\eta }^2}}+{\displaystyle \frac{{\displaystyle {\vartheta }_1^2}}{{\eta }^2}}+{\displaystyle \frac{24{\displaystyle {\vartheta }_1^2}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{6{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}-{\displaystyle \frac{24{\vartheta }_1{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{48{\vartheta }_1{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{4{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{6{\vartheta }_1{z}^2}{{\eta }^2}}+{\displaystyle \frac{72{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{4{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{24{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{z^4}{{\eta }^2}}\hfill \\ +{\displaystyle \frac{4z^3}{{\eta }^2}}+{\displaystyle \frac{2z^2}{{\eta }^2}}\hfill \end{array}$
5	$\begin{array}{c}{\displaystyle \frac{{\displaystyle {\vartheta }_1^5}}{{\eta }^2}}-{\displaystyle \frac{5{\displaystyle {\vartheta }_1^4}}{2{\eta }^2}}+{\displaystyle \frac{5{\displaystyle {\vartheta }_1^4}z}{{\eta }^2}}+{\displaystyle \frac{5{\displaystyle {\vartheta }_1^3}}{3{\eta }^2}}+{\displaystyle \frac{40{\displaystyle {\vartheta }_1^3}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{10{\displaystyle {\vartheta }_1^3}{z}^2}{{\eta }^2}}-{\displaystyle \frac{60{\displaystyle {\vartheta }_1^2}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{120{\displaystyle {\vartheta }_1^2}{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{10{\displaystyle {\vartheta }_1^2}{z}^3}{{\eta }^2}}+{\displaystyle \frac{15{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}-{\displaystyle \frac{{\vartheta }_1}{6{\eta }^2}}+{\displaystyle \frac{360{\vartheta }_1{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{20{\vartheta }_1{\vartheta }_2}{{\eta }^3}}\hfill \\ +{\displaystyle \frac{120{\vartheta }_1{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{5{\vartheta }_1{z}^4}{{\eta }^2}}+{\displaystyle \frac{20{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{10{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{180{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{360{\displaystyle {\vartheta }_2^2}z}{{\eta }^4}}+{\displaystyle \frac{40{\vartheta }_2{z}^3}{{\eta }^3}}+{\displaystyle \frac{60{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{z^5}{{\eta }^2}}+{\displaystyle \frac{15z^4}{2{\eta }^2}}+{\displaystyle \frac{35z^3}{3{\eta }^2}}+{\displaystyle \frac{5z^2}{2{\eta }^2}}\hfill \end{array}$
6	$\begin{array}{c}{\displaystyle \frac{1}{42{\eta }^2}}+{\displaystyle \frac{{\displaystyle {\vartheta }_1^6}}{{\eta }^2}}-{\displaystyle \frac{3{\displaystyle {\vartheta }_1^5}}{{\eta }^2}}+{\displaystyle \frac{6{\displaystyle {\vartheta }_1^5}z}{{\eta }^2}}+{\displaystyle \frac{5{\displaystyle {\vartheta }_1^4}}{2{\eta }^2}}+{\displaystyle \frac{60{\displaystyle {\vartheta }_1^4}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{15{\displaystyle {\vartheta }_1^4}{z}^2}{{\eta }^2}}-{\displaystyle \frac{120{\displaystyle {\vartheta }_1^3}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{240{\displaystyle {\vartheta }_1^3}{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{20{\displaystyle {\vartheta }_1^3}{z}^3}{{\eta }^2}}+{\displaystyle \frac{30{\displaystyle {\vartheta }_1^3}{z}^2}{{\eta }^2}}-{\displaystyle \frac{{\displaystyle {\vartheta }_1^2}}{2{\eta }^2}}+{\displaystyle \frac{1080{\displaystyle {\vartheta }_1^2}{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}\hfill \\ +{\displaystyle \frac{60{\displaystyle {\vartheta }_1^2}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{360{\displaystyle {\vartheta }_1^2}{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{15{\displaystyle {\vartheta }_1^2}{z}^4}{{\eta }^2}}+{\displaystyle \frac{60{\displaystyle {\vartheta }_1^2}{z}^3}{{\eta }^2}}+{\displaystyle \frac{30{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}-{\displaystyle \frac{1080{\vartheta }_1{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{2160{\vartheta }_1{\displaystyle {\vartheta }_2^2}z}{{\eta }^4}}+{\displaystyle \frac{240{\vartheta }_1{\vartheta }_2{z}^3}{{\eta }^3}}+{\displaystyle \frac{360{\vartheta }_1{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{6{\vartheta }_1{z}^5}{{\eta }^2}}+{\displaystyle \frac{45{\vartheta }_1{z}^4}{{\eta }^2}}\hfill \\ +{\displaystyle \frac{70{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{15{\vartheta }_1{z}^2}{{\eta }^2}}+{\displaystyle \frac{2880{\displaystyle {\vartheta }_2^3}}{{\eta }^5}}+{\displaystyle \frac{180{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{1080{\displaystyle {\vartheta }_2^2}{z}^2}{{\eta }^4}}-{\displaystyle \frac{2{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{60{\vartheta }_2{z}^4}{{\eta }^3}}+{\displaystyle \frac{240{\vartheta }_2{z}^3}{{\eta }^3}}+{\displaystyle \frac{120{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{z^6}{{\eta }^2}}+{\displaystyle \frac{12z^5}{{\eta }^2}}+{\displaystyle \frac{75z^4}{2{\eta }^2}}+{\displaystyle \frac{30z^3}{{\eta }^2}}+{\displaystyle \frac{3z^2}{{\eta }^2}}\hfill \end{array}$

.

6.2. Fractional Gould–Hopper–Bell–Apostol–Euler Polynomials

For $\alpha =0,a=-1$, and $b=1$, UATP ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\alpha ,\zeta ,a,b)$ reduce to the Euler polynomials ${\displaystyle {\mathcal{E}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta )$, i.e., ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;0,\zeta ,-1,1)={\displaystyle {\mathcal{E}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta )$. Consequently, for the same selection of $\alpha ,a$ and b FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ reduce to FGHBlAEP ${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$, which are defined by:

$${\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)}^{-\sigma }\left\{{}_{\mathcal{B}el}{\displaystyle {\mathbb{E}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta )\right\}={\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ).$$

(76)

The generating function for FGHBlAEP ${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ is given as:

$${\left({\displaystyle \frac{2}{\zeta e^{\omega }+1}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(77)

Some corresponding results for FGHBlAEP ${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ are presented in

Table 3. Results for FGHBlAEP ${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$.

Series	${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )=\frac{\nu !}{{\eta }^{\sigma }}{\displaystyle {\sum }_{\kappa =0}^{\left[\frac{\nu }{r}\right]}}\frac{{\left(\sigma \right)}_{\kappa }{\displaystyle {\vartheta }_2^{\kappa }}{}_{\mathcal{B}el}{\displaystyle {\mathcal{E}}_{\nu -rk}^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta )}{{\eta }^{\kappa }\kappa !(\nu -r\kappa )!}$
representations	${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathcal{E}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta )\mathcal{B}e{l}_{\kappa }\left(z\right)$
	${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathcal{E}}_{\nu -\kappa }^{\left(\epsilon \right)}}\left(\zeta \right){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )$
Summation	${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z+\rho ;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{B}el}{\displaystyle {\mathcal{E}}_{\nu -\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}(h,{\vartheta }_2,\rho ;\eta )$
formulas	${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon +\iota ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathcal{E}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta ){}_{\mathcal{B}el}{\displaystyle {\mathcal{E}}_{\kappa }^{(\iota ,r)}}(z;\zeta )$
	${\displaystyle {\mathbb{E}}_{\nu ,\sigma +\upsilon }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathcal{E}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta ){}_{\mathcal{H}}{\displaystyle {\mathcal{E}}_{\kappa ,\upsilon }^{\left(r\right)}}(h,{\vartheta }_2,z;\eta )$
	${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle \frac{\zeta }{2}}{\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\mathcal{E}}_{\nu -\kappa }\left(\zeta \right)\left\{{\displaystyle {\mathbb{E}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+1,{\vartheta }_2,z;\zeta ,\eta )+{\displaystyle {\mathbb{E}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )\right\}$
Symmetry	${\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)p^{\nu -\kappa }{q}^{\kappa }{\displaystyle {\mathbb{E}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{E}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{\displaystyle {\mathbb{E}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{E}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
formulas	${\displaystyle {\sum }_{\kappa =0}^{\nu }}{\displaystyle {\sum }_{\varpi =0}^{q-1}}{\displaystyle {\sum }_{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){(-\zeta )}^{(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{(-1)}^{(p+q-2)}{\displaystyle {\mathbb{E}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2,z;\zeta \eta ){\displaystyle {\mathbb{E}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle {\sum }_{\kappa =0}^{\nu }}{\displaystyle {\sum }_{\varpi =0}^{p-1}}{\displaystyle {\sum }_{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){(-\zeta )}^{(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{(-1)}^{(p+q-2)}{\displaystyle {\mathbb{E}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{E}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\zeta ,\eta )$
Multiplicative and	${\widehat{M}}_{\mathbb{E}}={\vartheta }_1+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^{r-1}}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}-{\displaystyle \frac{\epsilon \zeta e^{D_{{\vartheta }_1}}}{\zeta e^{D_{{\vartheta }_1}}+1}}$
derivative operators	${\widehat{P}}_{\mathbb{E}}=D_{\vartheta 1}$
Differential equation	$\left({\vartheta }_1{D}_{{\vartheta }_1}+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^r}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}{D}_{{\vartheta }_1}-{\displaystyle \frac{\epsilon \zeta e^{D_{{\vartheta }_1}}}{\zeta e^{D_{{\vartheta }_1}}+1}}{D}_{{\vartheta }_1}-\nu \right){\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )=0$

.

The first seven FGHBlAEP ${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ for $\zeta =\epsilon =1,r=\sigma =2$ are listed in

Table 4. The first seven FGHBlAEP ${\displaystyle {\mathbb{E}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$.

$\mathit{\nu }$	${\displaystyle {\mathbb{E}}_{\mathit{\nu },\mathit{\sigma }}^{(\mathit{\epsilon },\mathit{r})}}({\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2,\mathit{z};\mathit{\zeta },\mathit{\eta })$
0	${\displaystyle \frac{1}{{\eta }^2}}$
1	$-{\displaystyle \frac{1}{2{\eta }^2}}+{\displaystyle \frac{{\vartheta }_1}{{\eta }^2}}+{\displaystyle \frac{z}{{\eta }^2}}$
2	${\displaystyle \frac{{\displaystyle {\vartheta }_1^2}}{{\eta }^2}}-{\displaystyle \frac{{\vartheta }_1}{{\eta }^2}}+{\displaystyle \frac{2{\vartheta }_{1}z}{{\eta }^2}}+{\displaystyle \frac{4{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{z^2}{{\eta }^2}}$
3	${\displaystyle \frac{1}{4{\eta }^2}}+{\displaystyle \frac{{\displaystyle {\vartheta }_1^3}}{{\eta }^2}}-{\displaystyle \frac{3{\displaystyle {\vartheta }_1^2}}{2{\eta }^2}}+{\displaystyle \frac{3{\displaystyle {\vartheta }_1^2}z}{{\eta }^2}}+{\displaystyle \frac{12{\vartheta }_1{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{3{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{6{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{12{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{z^3}{{\eta }^2}}+{\displaystyle \frac{3z^2}{2{\eta }^2}}-{\displaystyle \frac{z}{2{\eta }^2}}$
4	$\begin{array}{c}{\displaystyle \frac{{\displaystyle {\vartheta }_1^4}}{{\eta }^2}}-{\displaystyle \frac{2{\displaystyle {\vartheta }_1^3}}{{\eta }^2}}+{\displaystyle \frac{4{\displaystyle {\vartheta }_1^3}z}{{\eta }^2}}+{\displaystyle \frac{24{\displaystyle {\vartheta }_1^2}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{6{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}+{\displaystyle \frac{{\vartheta }_1}{{\eta }^2}}-{\displaystyle \frac{24{\vartheta }_1{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{48{\vartheta }_1{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{4{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{6{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{2{\vartheta }_{1}z}{{\eta }^2}}+{\displaystyle \frac{72{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{24{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{z^4}{{\eta }^2}}\hfill \\ +{\displaystyle \frac{4z^3}{{\eta }^2}}+{\displaystyle \frac{z^2}{{\eta }^2}}\hfill \end{array}$
5	$\begin{array}{c}-{\displaystyle \frac{1}{2{\eta }^2}}+{\displaystyle \frac{{\displaystyle {\vartheta }_1^5}}{{\eta }^2}}-{\displaystyle \frac{5{\displaystyle {\vartheta }_1^4}}{2{\eta }^2}}+{\displaystyle \frac{5{\displaystyle {\vartheta }_1^4}z}{{\eta }^2}}+{\displaystyle \frac{40{\displaystyle {\vartheta }_1^3}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{10{\displaystyle {\vartheta }_1^3}{z}^2}{{\eta }^2}}+{\displaystyle \frac{5{\displaystyle {\vartheta }_1^2}}{2{\eta }^2}}-{\displaystyle \frac{60{\displaystyle {\vartheta }_1^2}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{120{\displaystyle {\vartheta }_1^2}{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{10{\displaystyle {\vartheta }_1^2}{z}^3}{{\eta }^2}}+{\displaystyle \frac{15{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}-{\displaystyle \frac{5{\displaystyle {\vartheta }_1^2}z}{{\eta }^2}}+{\displaystyle \frac{360{\vartheta }_1{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{120{\vartheta }_1{\vartheta }_2{z}^2}{{\eta }^3}}\hfill \\ +{\displaystyle \frac{5{\vartheta }_1{z}^4}{{\eta }^2}}+{\displaystyle \frac{20{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{5{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{180{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{360{\displaystyle {\vartheta }_2^2}z}{{\eta }^4}}+{\displaystyle \frac{10{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{40{\vartheta }_2{z}^3}{{\eta }^3}}+{\displaystyle \frac{60{\vartheta }_2{z}^2}{{\eta }^3}}-{\displaystyle \frac{20{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{z^5}{{\eta }^2}}+{\displaystyle \frac{15z^4}{2{\eta }^2}}+{\displaystyle \frac{10z^3}{{\eta }^2}}+{\displaystyle \frac{z}{{\eta }^2}}\hfill \end{array}$
6	$\begin{array}{c}{\displaystyle \frac{{\displaystyle {\vartheta }_1^6}}{{\eta }^2}}-{\displaystyle \frac{3{\displaystyle {\vartheta }_1^5}}{{\eta }^2}}+{\displaystyle \frac{6{\displaystyle {\vartheta }_1^5}z}{{\eta }^2}}+{\displaystyle \frac{60{\displaystyle {\vartheta }_1^4}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{15{\displaystyle {\vartheta }_1^4}{z}^2}{{\eta }^2}}+{\displaystyle \frac{5{\displaystyle {\vartheta }_1^3}}{{\eta }^2}}-{\displaystyle \frac{120{\displaystyle {\vartheta }_1^3}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{240{\displaystyle {\vartheta }_1^3}{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{20{\displaystyle {\vartheta }_1^3}{z}^3}{{\eta }^2}}+{\displaystyle \frac{30{\displaystyle {\vartheta }_1^3}{z}^2}{{\eta }^2}}-{\displaystyle \frac{10{\displaystyle {\vartheta }_1^3}z}{{\eta }^2}}+{\displaystyle \frac{1080{\displaystyle {\vartheta }_1^2}{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{360{\displaystyle {\vartheta }_1^2}{\vartheta }_2{z}^2}{{\eta }^3}}\hfill \\ +{\displaystyle \frac{15{\displaystyle {\vartheta }_1^2}{z}^4}{{\eta }^2}}+{\displaystyle \frac{60{\displaystyle {\vartheta }_1^2}{z}^3}{{\eta }^2}}+{\displaystyle \frac{15{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}-{\displaystyle \frac{3{\vartheta }_1}{{\eta }^2}}-{\displaystyle \frac{1080{\vartheta }_1{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{2160{\vartheta }_1{\displaystyle {\vartheta }_2^2}z}{{\eta }^4}}+{\displaystyle \frac{60{\vartheta }_1{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{240{\vartheta }_1{\vartheta }_2{z}^3}{{\eta }^3}}+{\displaystyle \frac{360{\vartheta }_1{\vartheta }_2{z}^2}{{\eta }^3}}-{\displaystyle \frac{120{\vartheta }_1{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{6{\vartheta }_1{z}^5}{{\eta }^2}}\hfill \\ +{\displaystyle \frac{45{\vartheta }_1{z}^4}{{\eta }^2}}+{\displaystyle \frac{60{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{6{\vartheta }_{1}z}{{\eta }^2}}+{\displaystyle \frac{2880{\displaystyle {\vartheta }_2^3}}{{\eta }^5}}+{\displaystyle \frac{1080{\displaystyle {\vartheta }_2^2}{z}^2}{{\eta }^4}}+{\displaystyle \frac{60{\vartheta }_2{z}^4}{{\eta }^3}}+{\displaystyle \frac{240{\vartheta }_2{z}^3}{{\eta }^3}}+{\displaystyle \frac{60{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{z^6}{{\eta }^2}}+{\displaystyle \frac{12z^5}{{\eta }^2}}+{\displaystyle \frac{35z^4}{{\eta }^2}}+{\displaystyle \frac{20z^3}{{\eta }^2}}+{\displaystyle \frac{z^2}{{\eta }^2}}\hfill \end{array}$

.

6.3. Fractional Gould–Hopper–Bell–Apostol–Genocchi Polynomials

For $\alpha =1,a=-{\displaystyle \frac{1}{2}},b=1$, and $\zeta ⟶{\displaystyle \frac{\zeta }{2}}$, UATP ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\alpha ,\zeta ,a,b)$ reduce to the Genocchi polynomials ${\displaystyle {\mathcal{G}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta )$, i.e., ${\displaystyle {\mathsf{P}}_{\nu }^{\left(\epsilon \right)}}\left(\vartheta ;1,{\displaystyle \frac{\zeta }{2}},-{\displaystyle \frac{1}{2}},1\right)={\displaystyle {\mathcal{G}}_{\nu }^{\left(\epsilon \right)}}(\vartheta ;\zeta )$. Consequently, for the same selection of $\alpha ,a,b$ and $\zeta$ FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ reduce to FGHBlAGP ${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$, which are defined by:

$${\left(\eta -{\vartheta }_2{\displaystyle \frac{{\partial }^r}{\partial {\displaystyle {\vartheta }_1^r}}}\right)}^{-\sigma }\left\{{}_{\mathcal{B}el}{\displaystyle {\mathbb{G}}_{\nu }^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta )\right\}={\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ).$$

(78)

The generating function for FGHBlAGP ${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ is given as:

$${\left({\displaystyle \frac{2\omega }{\zeta e^{\omega }+1}}\right)}^{\epsilon }{\displaystyle \frac{e^{{\vartheta }_1\omega +z(e^{\omega }-1)}}{{(\eta -{\vartheta }_2{\omega }^r)}^{\sigma }}}={\displaystyle \sum _{\nu =0}^{\infty }}{\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle \frac{{\omega }^{\nu }}{\nu !}}.$$

(79)

Certain corresponding results for FGHBlAGP ${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ are shown in

Table 5. Results for FGHBlAGP ${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$.

Series	${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )=\frac{\nu !}{{\eta }^{\sigma }}{\displaystyle {\sum }_{\kappa =0}^{\left[\frac{\nu }{r}\right]}}\frac{{\left(\sigma \right)}_{\kappa }{\displaystyle {\vartheta }_2^{\kappa }}{}_{\mathcal{B}el}{\displaystyle {\mathcal{G}}_{\nu -rk}^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta )}{{\eta }^{\kappa }\kappa !(\nu -r\kappa )!}$
representations	${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathcal{G}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta )\mathcal{B}e{l}_{\kappa }\left(z\right)$
	${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\displaystyle {\mathcal{G}}_{\nu -\kappa }^{\left(\epsilon \right)}}\left(\zeta \right){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}({\vartheta }_1,{\vartheta }_2,z;\eta )$
Summation	${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z+\rho ;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{B}el}{\displaystyle {\mathcal{G}}_{\nu -\kappa }^{\left(\epsilon \right)}}({\vartheta }_1,z;\zeta ){}_{\mathcal{H}}\mathcal{B}e{\displaystyle l_{\kappa ,\sigma }^{\left(r\right)}}(h,{\vartheta }_2,\rho ;\eta )$
formulas	${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon +\iota ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathcal{G}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta ){}_{\mathcal{B}el}{\displaystyle {\mathcal{G}}_{\kappa }^{(\iota ,r)}}(z;\zeta )$
	${\displaystyle {\mathbb{G}}_{\nu ,\sigma +\upsilon }^{(\epsilon ,r)}}({\vartheta }_1+h,{\vartheta }_2,z;\zeta ,\eta )={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){}_{\mathcal{H}}{\displaystyle {\mathcal{G}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2;\zeta ,\eta ){}_{\mathcal{H}}{\displaystyle {\mathcal{E}}_{\kappa ,\upsilon }^{\left(r\right)}}(h,{\vartheta }_2,z;\eta )$
	${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle \frac{\zeta }{4}}{\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){\mathcal{E}}_{\nu -\kappa }\left(\zeta \right)\left\{{\displaystyle {\mathbb{G}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1+1,{\vartheta }_2,z;\zeta ,\eta )+{\displaystyle {\mathbb{G}}_{\kappa ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )\right\}$
Symmetry	${\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)p^{\nu -\kappa }{q}^{\kappa }{\displaystyle {\mathbb{G}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{G}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle {\sum }_{\kappa =0}^{\nu }}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right)q^{\nu -\kappa }{p}^{\kappa }{\displaystyle {\mathbb{G}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{G}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$
formulas	${\displaystyle {\sum }_{\kappa =0}^{\nu }}{\displaystyle {\sum }_{\varpi =0}^{q-1}}{\displaystyle {\sum }_{\varsigma =0}^{p-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){(-\zeta )}^{(\varpi +\varsigma )}{p}^{\nu -\kappa }{q}^{\kappa }{(-{\displaystyle \frac{1}{2}})}^{(p+q-2)}{\displaystyle {\mathbb{G}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varpi ,{\vartheta }_2,z;\zeta \eta ){\displaystyle {\mathbb{G}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\zeta ,\eta )$
	$={\displaystyle {\sum }_{\kappa =0}^{\nu }}{\displaystyle {\sum }_{\varpi =0}^{p-1}}{\displaystyle {\sum }_{\varsigma =0}^{q-1}}\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{\kappa }}\right){(-\zeta )}^{(\varpi +\varsigma )}{q}^{\nu -\kappa }{p}^{\kappa }{(-{\displaystyle \frac{1}{2}})}^{(p+q-2)}{\displaystyle {\mathbb{G}}_{\nu -\kappa ,\sigma }^{(\epsilon ,r)}}(p{\vartheta }_1+\varpi ,{\vartheta }_2,z;\zeta ,\eta ){\displaystyle {\mathbb{G}}_{\kappa ,\sigma }^{(\epsilon ,r)}}(q{\vartheta }_1+\varsigma ,{\vartheta }_2,z;\zeta ,\eta )$
Multiplicative and	${\widehat{M}}_{\mathbb{G}}={\vartheta }_1+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^{r-1}}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}+{\displaystyle \frac{\epsilon }{D_{{\vartheta }_1}}}-{\displaystyle \frac{\epsilon \zeta e^{D_{{\vartheta }_1}}}{\zeta e^{D_{{\vartheta }_1}}+1}}$
derivative operators	${\widehat{P}}_{\mathbb{G}}=D_{\vartheta 1}$
Differential equation	$\left(\epsilon +{\vartheta }_1{D}_{{\vartheta }_1}+{\displaystyle \frac{\sigma {\vartheta }_{2}r{\displaystyle D_{{\vartheta }_1}^r}}{\eta -{\vartheta }_2{\displaystyle D_{{\vartheta }_1}^r}}}+z{e}^{D_{{\vartheta }_1}}{D}_{{\vartheta }_1}-{\displaystyle \frac{\epsilon \zeta e^{D_{{\vartheta }_1}}}{\zeta e^{D_{{\vartheta }_1}}+1}}{D}_{{\vartheta }_1}-\nu \right){\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )=0$

.

The first seven FGHBlAGP ${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$ for $\zeta =\epsilon =1,r=\sigma =2$ are listed in

Table 6. The first seven FGHBlAGP ${\displaystyle {\mathbb{G}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\zeta ,\eta )$.

$\mathit{\nu }$	${\displaystyle {\mathbb{G}}_{\mathit{\nu },\mathit{\sigma }}^{(\mathit{\epsilon },\mathit{r})}}({\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2,\mathit{z};\mathit{\zeta },\mathit{\eta })$
0	0
1	${\displaystyle \frac{1}{{\eta }^2}}$
2	$-{\displaystyle \frac{1}{{\eta }^2}}+{\displaystyle \frac{2{\vartheta }_1}{{\eta }^2}}+{\displaystyle \frac{2z}{{\eta }^2}}$
3	${\displaystyle \frac{3{\displaystyle {\vartheta }_1^2}}{{\eta }^2}}-{\displaystyle \frac{3{\vartheta }_1}{{\eta }^2}}+{\displaystyle \frac{6{\vartheta }_{1}z}{{\eta }^2}}+{\displaystyle \frac{12{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{3z^2}{{\eta }^2}}$
4	$\begin{array}{c}{\displaystyle \frac{1}{{\eta }^2}}+{\displaystyle \frac{4{\displaystyle {\vartheta }_1^3}}{{\eta }^2}}-{\displaystyle \frac{6{\displaystyle {\vartheta }_1^2}}{{\eta }^2}}+{\displaystyle \frac{12{\displaystyle {\vartheta }_1^2}z}{{\eta }^2}}+{\displaystyle \frac{48{\vartheta }_1{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{12{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{24{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{48{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{4z^3}{{\eta }^2}}+{\displaystyle \frac{6z^2}{{\eta }^2}}-{\displaystyle \frac{2z}{{\eta }^2}}\hfill \end{array}$
5	$\begin{array}{c}{\displaystyle \frac{5{\displaystyle {\vartheta }_1^4}}{{\eta }^2}}-{\displaystyle \frac{10{\displaystyle {\vartheta }_1^3}}{{\eta }^2}}+{\displaystyle \frac{20{\displaystyle {\vartheta }_1^3}z}{{\eta }^2}}+{\displaystyle \frac{120{\displaystyle {\vartheta }_1^2}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{30{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}+{\displaystyle \frac{5{\vartheta }_1}{{\eta }^2}}-{\displaystyle \frac{120{\vartheta }_1{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{240{\vartheta }_1{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{20{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{30{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{10{\vartheta }_{1}z}{{\eta }^2}}+{\displaystyle \frac{360{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{120{\vartheta }_2{z}^2}{{\eta }^3}}+{\displaystyle \frac{5z^4}{{\eta }^2}}\hfill \\ +{\displaystyle \frac{20z^3}{{\eta }^2}}+{\displaystyle \frac{5z^2}{{\eta }^2}}\hfill \end{array}$
6	$\begin{array}{c}-{\displaystyle \frac{3}{{\eta }^2}}+{\displaystyle \frac{6{\displaystyle {\vartheta }_1^5}}{{\eta }^2}}-{\displaystyle \frac{15{\displaystyle {\vartheta }_1^4}}{{\eta }^2}}+{\displaystyle \frac{30{\displaystyle {\vartheta }_1^4}z}{{\eta }^2}}+{\displaystyle \frac{240{\displaystyle {\vartheta }_1^3}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{60{\displaystyle {\vartheta }_1^3}{z}^2}{{\eta }^2}}+{\displaystyle \frac{15{\displaystyle {\vartheta }_1^2}}{{\eta }^2}}-{\displaystyle \frac{360{\displaystyle {\vartheta }_1^2}{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{720{\displaystyle {\vartheta }_1^2}{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{60{\displaystyle {\vartheta }_1^2}{z}^3}{{\eta }^2}}+{\displaystyle \frac{90{\displaystyle {\vartheta }_1^2}{z}^2}{{\eta }^2}}-{\displaystyle \frac{30{\displaystyle {\vartheta }_1^2}z}{{\eta }^2}}+{\displaystyle \frac{2160{\vartheta }_1{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{720{\vartheta }_1{\vartheta }_2{z}^2}{{\eta }^3}}\hfill \\ +{\displaystyle \frac{30{\vartheta }_1{z}^4}{{\eta }^2}}+{\displaystyle \frac{120{\vartheta }_1{z}^3}{{\eta }^2}}+{\displaystyle \frac{30{\vartheta }_1{z}^2}{{\eta }^2}}-{\displaystyle \frac{1080{\displaystyle {\vartheta }_2^2}}{{\eta }^4}}+{\displaystyle \frac{2160{\displaystyle {\vartheta }_2^2}z}{{\eta }^4}}+{\displaystyle \frac{60{\vartheta }_2}{{\eta }^3}}+{\displaystyle \frac{240{\vartheta }_2{z}^3}{{\eta }^3}}+{\displaystyle \frac{360{\vartheta }_2{z}^2}{{\eta }^3}}-{\displaystyle \frac{120{\vartheta }_{2}z}{{\eta }^3}}+{\displaystyle \frac{6z^5}{{\eta }^2}}+{\displaystyle \frac{45z^4}{{\eta }^2}}+{\displaystyle \frac{60z^3}{{\eta }^2}}+{\displaystyle \frac{6z}{{\eta }^2}}\hfill \end{array}$

.

7. Approximate Zeros and Graphical Representations

This section examines zero distributions and provides graphical demonstrations of FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ for specific parameter values and indices.

Here, we start by providing the first seven terms of FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ for $\alpha =\zeta =a=b=1$ and $r=\epsilon =2$ which are listed in

Table 7. The first seven FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )$ for $\nu =\zeta =a=b=1$ and $r=\epsilon =2$.

$\mathit{\nu }$	${\displaystyle {\mathbb{P}}_{\mathit{\nu },\mathit{\sigma }}^{(\mathit{\epsilon },\mathit{r})}}({\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2,\mathit{z};\mathit{\alpha },\mathit{\zeta },\mathit{a},\mathit{b},\mathit{\eta })$
0	${\eta }^{-\sigma }$
1	${\eta }^{-\sigma }({\vartheta }_1+z-1)$
2	${\displaystyle \frac{1}{6}}\left(5{\eta }^{-\sigma }+6{\displaystyle {\vartheta }_1^2}{\eta }^{-\sigma }-12{\vartheta }_1{\eta }^{-\sigma }+12{\vartheta }_{1}z{\eta }^{-\sigma }+12\sigma {\vartheta }_2{\eta }^{-\sigma -1}+6z^2{\eta }^{-\sigma }-6z{\eta }^{-\sigma }\right)$
3	$\begin{array}{c}{\displaystyle \frac{1}{2}}\left(-{\eta }^{-\sigma }+2{\displaystyle {\vartheta }_1^3}{\eta }^{-\sigma }-6{\displaystyle {\vartheta }_1^2}{\eta }^{-\sigma }+6{\displaystyle {\vartheta }_1^2}z{\eta }^{-\sigma }+5{\vartheta }_1{\eta }^{-\sigma }+12\sigma {\vartheta }_1{\vartheta }_2{\eta }^{-\sigma -1}+6{\vartheta }_1{z}^2{\eta }^{-\sigma }-6{\vartheta }_{1}z{\eta }^{-\sigma }-12\sigma {\vartheta }_2{\eta }^{-\sigma -1}\right.\hfill \\ \left.+12\sigma {\vartheta }_{2}z{\eta }^{-\sigma -1}+2z^3{\eta }^{-\sigma }+z{\eta }^{-\sigma }\right)\hfill \end{array}$
4	$\begin{array}{c}{\displaystyle \frac{1}{10}}\left({\eta }^{-\sigma }+10{\displaystyle {\vartheta }_1^4}{\eta }^{-\sigma }-40{\displaystyle {\vartheta }_1^3}{\eta }^{-\sigma }+40{\displaystyle {\vartheta }_1^3}z{\eta }^{-\sigma }+50{\displaystyle {\vartheta }_1^2}{\eta }^{-\sigma }+120\sigma {\displaystyle {\vartheta }_1^2}{\vartheta }_2{\eta }^{-\sigma -1}+60{\displaystyle {\vartheta }_1^2}{z}^2{\eta }^{-\sigma }-60{\displaystyle {\vartheta }_1^2}z{\eta }^{-\sigma }-20{\vartheta }_1{\eta }^{-\sigma }\right.\hfill \\ -240\sigma {\vartheta }_1{\vartheta }_2{\eta }^{-\sigma -1}+240\sigma {\vartheta }_1{\vartheta }_{2}z{\eta }^{-\sigma -1}+40{\vartheta }_1{z}^3{\eta }^{-\sigma }+20{\vartheta }_{1}z{\eta }^{-\sigma }+120{\sigma }^2{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}+120\sigma {\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}+100\sigma {\vartheta }_2{\eta }^{-\sigma -1}\hfill \\ \left.+120\sigma {\vartheta }_2{z}^2{\eta }^{-\sigma -1}-120\sigma {\vartheta }_{2}z{\eta }^{-\sigma -1}+10z^4{\eta }^{-\sigma }+20z^3{\eta }^{-\sigma }\right)\hfill \end{array}$
5	$\begin{array}{c}{\displaystyle \frac{1}{6}}\left({\eta }^{-\sigma }+6{\displaystyle {\vartheta }_1^5}{\eta }^{-\sigma }-30{\displaystyle {\vartheta }_1^4}{\eta }^{-\sigma }+30{\displaystyle {\vartheta }_1^4}z{\eta }^{-\sigma }+50{\displaystyle {\vartheta }_1^3}{\eta }^{-\sigma }+120\sigma {\displaystyle {\vartheta }_1^3}{\vartheta }_2{\eta }^{-\sigma -1}+60{\displaystyle {\vartheta }_1^3}{z}^2{\eta }^{-\sigma }-60{\displaystyle {\vartheta }_1^3}z{\eta }^{-\sigma }-30{\displaystyle {\vartheta }_1^2}{\eta }^{-\sigma }\right.\hfill \\ -360\sigma {\displaystyle {\vartheta }_1^2}{\vartheta }_2{\eta }^{-\sigma -1}+360\sigma {\displaystyle {\vartheta }_1^2}{\vartheta }_{2}z{\eta }^{-\sigma -1}+60{\displaystyle {\vartheta }_1^2}{z}^3{\eta }^{-\sigma }+30{\displaystyle {\vartheta }_1^2}z{\eta }^{-\sigma }+3{\vartheta }_1{\eta }^{-\sigma }+360{\sigma }^2{\vartheta }_1{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}+360\sigma {\vartheta }_1{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}\hfill \\ +300\sigma {\vartheta }_1{\vartheta }_2{\eta }^{-\sigma -1}+360\sigma {\vartheta }_1{\vartheta }_2{z}^2{\eta }^{-\sigma -1}-360\sigma {\vartheta }_1{\vartheta }_{2}z{\eta }^{-\sigma -1}+30{\vartheta }_1{z}^4{\eta }^{-\sigma }+60{\vartheta }_1{z}^3{\eta }^{-\sigma }-360{\sigma }^2{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}-360\sigma {\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}\hfill \\ +360{\sigma }^2{\displaystyle {\vartheta }_2^2}z{\eta }^{-\sigma -2}+360\sigma {\displaystyle {\vartheta }_2^2}z{\eta }^{-\sigma -2}-60\sigma {\vartheta }_2{\eta }^{-\sigma -1}+120\sigma {\vartheta }_2{z}^3{\eta }^{-\sigma -1}+60\sigma {\vartheta }_{2}z{\eta }^{-\sigma -1}+6z^5{\eta }^{-\sigma }+30z^4{\eta }^{-\sigma }+20z^3{\eta }^{-\sigma }\hfill \\ \left.-z{\eta }^{-\sigma }\right)\hfill \end{array}$
6	$\begin{array}{c}{\displaystyle \frac{1}{42}}\left(-5{\eta }^{-\sigma }+42{\displaystyle {\vartheta }_1^6}{\eta }^{-\sigma }-252{\displaystyle {\vartheta }_1^5}{\eta }^{-\sigma }+252{\displaystyle {\vartheta }_1^5}z{\eta }^{-\sigma }+525{\displaystyle {\vartheta }_1^4}{\eta }^{-\sigma }+1260\sigma {\displaystyle {\vartheta }_1^4}{\vartheta }_2{\eta }^{-\sigma -1}+630{\displaystyle {\vartheta }_1^4}{z}^2{\eta }^{-\sigma }-630{\displaystyle {\vartheta }_1^4}z{\eta }^{-\sigma }\right.\hfill \\ -420{\displaystyle {\vartheta }_1^3}{\eta }^{-\sigma }-5040\sigma {\displaystyle {\vartheta }_1^3}{\vartheta }_2{\eta }^{-\sigma -1}+5040\sigma {\displaystyle {\vartheta }_1^3}{\vartheta }_{2}z{\eta }^{-\sigma -1}+840{\displaystyle {\vartheta }_1^3}{z}^3{\eta }^{-\sigma }+420{\displaystyle {\vartheta }_1^3}z{\eta }^{-\sigma }+63{\displaystyle {\vartheta }_1^2}{\eta }^{-\sigma }+7560{\sigma }^2{\displaystyle {\vartheta }_1^2}{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}\hfill \\ +7560\sigma {\displaystyle {\vartheta }_1^2}{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}+6300\sigma {\displaystyle {\vartheta }_1^2}{\vartheta }_2{\eta }^{-\sigma -1}+7560\sigma {\displaystyle {\vartheta }_1^2}{\vartheta }_2{z}^2{\eta }^{-\sigma -1}-7560\sigma {\displaystyle {\vartheta }_1^2}{\vartheta }_{2}z{\eta }^{-\sigma -1}+630{\displaystyle {\vartheta }_1^2}{z}^4{\eta }^{-\sigma }+1260{\displaystyle {\vartheta }_1^2}{z}^3{\eta }^{-\sigma }\hfill \\ +42{\vartheta }_1{\eta }^{-\sigma }-15120{\sigma }^2{\vartheta }_1{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}-15120\sigma {\vartheta }_1{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}+15120{\sigma }^2{\vartheta }_1{\displaystyle {\vartheta }_2^2}z{\eta }^{-\sigma -2}+15120\sigma {\vartheta }_1{\displaystyle {\vartheta }_2^2}z{\eta }^{-\sigma -2}-2520\sigma {\vartheta }_1{\vartheta }_2{\eta }^{-\sigma -1}\hfill \\ +5040\sigma {\vartheta }_1{\vartheta }_2{z}^3{\eta }^{-\sigma -1}+2520\sigma {\vartheta }_1{\vartheta }_{2}z{\eta }^{-\sigma -1}+252{\vartheta }_1{z}^5{\eta }^{-\sigma }+1260{\vartheta }_1{z}^4{\eta }^{-\sigma }+840{\vartheta }_1{z}^3{\eta }^{-\sigma }-42{\vartheta }_{1}z{\eta }^{-\sigma }+5040{\sigma }^3{\displaystyle {\vartheta }_2^3}{\eta }^{-\sigma -3}\hfill \\ +15120{\sigma }^2{\displaystyle {\vartheta }_2^3}{\eta }^{-\sigma -3}+10080\sigma {\displaystyle {\vartheta }_2^3}{\eta }^{-\sigma -3}+6300{\sigma }^2{\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}+6300\sigma {\displaystyle {\vartheta }_2^2}{\eta }^{-\sigma -2}+7560{\sigma }^2{\displaystyle {\vartheta }_2^2}{z}^2{\eta }^{-\sigma -2}+7560\sigma {\displaystyle {\vartheta }_2^2}{z}^2{\eta }^{-\sigma -2}\hfill \\ -7560{\sigma }^2{\displaystyle {\vartheta }_2^2}z{\eta }^{-\sigma -2}-7560\sigma {\displaystyle {\vartheta }_2^2}z{\eta }^{-\sigma -2}+126\sigma {\vartheta }_2{\eta }^{-\sigma -1}+1260\sigma {\vartheta }_2{z}^4{\eta }^{-\sigma -1}+2520\sigma {\vartheta }_2{z}^3{\eta }^{-\sigma -1}+42z^6{\eta }^{-\sigma }+378z^5{\eta }^{-\sigma }\hfill \\ \left.+735z^4{\eta }^{-\sigma }+210z^3{\eta }^{-\sigma }\right)\hfill \end{array}$

.

Certain interesting zeros of FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )=0$ for $\alpha =\zeta =a=b=1$, $\sigma =\epsilon =2,\eta =5,{\vartheta }_2=z={\displaystyle \frac{1}{5}}$, $\nu =100$ and different values of r are shown in Figure 1.

The stacking configurations of the approximation zeros of FGHBlATP ${\displaystyle {\mathbb{P}}_{\nu ,\sigma }^{(\epsilon ,r)}}({\vartheta }_1,{\vartheta }_2,z;\alpha ,\zeta ,a,b,\eta )=0$ for $\alpha =\zeta =a=b=1$ and $\sigma =\epsilon =2,\eta =5$, ${\vartheta }_2=z={\displaystyle \frac{1}{5}},1\le \nu \le 100$ and different values of r yield 3D structures, as illustrated in Figure 2:

8. Conclusions

The combined application of special polynomials with integral transformations serves as a powerful instrument for formulating novel extensions of special polynomials. This study utilized the Euler integral to establish a novel class of special polynomials, known as the fractional Gould–Hopper–Bell–Apostol-type polynomials. By employing the operational technique and Euler integral in Section 2 and Section 3, we formulated the relevant operational definition, generating function, explicit expressions, and summation identities. Moreover, symmetry identities, multiplicative and derivative operators, and differential equations were formulated.

Notable members of this class, such as fractional Gould–Hopper–Bell, Apostol–Bernoulli, Euler, and Genocchi polynomials, were discussed, producing analogous results for each (Section 6). Furthermore, in Section 7, computational analyses utilizing Mathematica were performed to investigate the zero distributions and graphical representations of specific members of the fractional Gould–Hopper–Bell–Apostol-type polynomials. The visual and numerical outcomes enhance the understanding of the characteristics and behavior of these polynomials.

This study improves the applicability of the fractional operational technique to develop a generalized class of special polynomials, hence expanding and extending several recognized families of special polynomials. This study advances the field of special numbers and polynomial theory by establishing and analyzing a new, generalized class of special polynomials. Future research may explore the degenerate type of the class proposed in this study, along with their respective applications.