A Family of q-General Bell Polynomials: Construction, Properties and Applications

Abstract

This paper introduces a new family of q-special polynomials, termed q-general Bell polynomials, and systematically explores their structural and analytical properties. We establish their generating functions, derive explicit series representations, and develop recurrence relations to characterize their combinatorial behavior. Additionally, we characterize their quasi-monomial properties and construct associated differential equations governing these polynomials. To demonstrate the versatility and applicability of this family, we investigate certain examples, including the q-Gould–Hopper–Bell and q-truncated exponential-Bell polynomials, deriving analogous results for each. Further, we employ computational tools in Mathematica to examine zero distributions and produce visualizations, offering numerical and graphical insights into polynomial behavior.

1. Introduction

q-calculus represents both a generalization and an adaptation of classical calculus. The discipline first emerged in the 18th century [1,2], propelled by its utility across diverse areas of mathematics, physics, and engineering. Interest in q-series concepts with origins in the 1800s experienced a resurgence in the 1980s following breakthroughs in quantum groups, whose development and applications in modern mathematics and physics revitalized the study of this historically rich field. q-series, serving as an extension of hypergeometric series, have garnered significant attention in recent research. This surge in interest stems from their unexpected applications across diverse disciplines, including quantum groups, statistical mechanics, and transcendental number theory, highlighting their broad relevance in contemporary mathematics and physics.

A central focus in the study of q-calculus lies in q-special functions, which are intrinsic to the field and form a bridge between mathematics and physics. In the realm of mathematical physics, a variety of q-special polynomials and functions have been formulated and utilized to serve as representations within quantum algebra [3].

The fundamental principles and nomenclature of q-calculus have been extensively established in the works of Andrews, Gasper, Ernst, and others [4,5,6]. These q-analogues are pivotal for analyzing and extending classical special functions, thereby enriching the understanding of their properties in diverse contexts. In recent years, a growing body of research on q-special polynomials has further underscored their significance within both theoretical and applied mathematical fields.

Additionally, q-special polynomials such as q-Bernoulli, q-Euler, q-Appell, q-Sheffer–Appell, q-Fubini–Appell, and q-modified-Laguerre–Appell polynomials have been extensively investigated, with their properties and applications documented in numerous studies over time [7,8,9,10,11,12].

The q-shifted factorial ${(\gamma ;q)}_{\omega }$ is given as follows [6]:

$${(\gamma ;q)}_{\omega }=\prod _{\varrho =1}^{\omega }(1-q^{\varrho -1}\gamma ),\omega \in \mathbb{N};{(\gamma ;q)}_{\infty }=\prod _{\varrho =0}^{\infty }(1-q^{\varrho }\gamma ),\gamma \in \mathbb{C}.$$

(1)

The q-number for $\gamma \in \mathbb{N}$ and the q-factorial function [6] are, respectively, defined by

$${\left[\gamma \right]}_q={\displaystyle \frac{1-q^{\gamma }}{1-q}},(q\in \mathbb{C}\setminus \left\{1\right\})$$

(2)

and

$${\left[\gamma \right]}_q!=\prod _{\alpha =1}^{\gamma }{\left[\alpha \right]}_q={\left[1\right]}_q{\left[2\right]}_q{\left[3\right]}_q\dots {\left[\gamma \right]}_q={\displaystyle \frac{{(q;q)}_{\gamma }}{(1-q)\gamma }},{\left[0\right]}_q!=1,\gamma \in \mathbb{N},0<q<1.$$

(3)

The q-binomial coefficient ${\left[{\displaystyle \genfrac{}{}{0pt}{}{\gamma }{\alpha }}\right]}_q$ is specified by [6]

$${\left[{\displaystyle \genfrac{}{}{0pt}{}{\gamma }{\alpha }}\right]}_q={\displaystyle \frac{{\left[\gamma \right]}_q!}{{\left[\alpha \right]}_q!{[\gamma -\alpha ]}_q!}},\alpha =0,1,2,\dots ,\gamma ;\gamma \in {\mathbb{N}}_0,\alpha \le \gamma .$$

(4)

The q-derivative for a function $\mathcal{X}$ at a point $\vartheta \in \mathbb{C}-\left\{0\right\}$ is defined as follows [6]:

$${\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right)={\displaystyle \frac{\mathcal{X}\left(\vartheta \right)-\mathcal{X}\left(q\vartheta \right)}{\vartheta -q\vartheta }},0<q<1.$$

(5)

It is also worth noting that

$$\begin{array}{cc}\left(\mathrm{i}\right)& \underset{q\to 1}{lim}{\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right)={\displaystyle \frac{d\mathcal{X}\left(\vartheta \right)}{d\vartheta }},\mathrm{where}{\displaystyle \frac{d}{d\vartheta }}\mathrm{denotes}\mathrm{the}\mathrm{classical}\mathrm{ordinary}\mathrm{derivative},\hfill \\ \left(\mathrm{ii}\right)& {\mathfrak{D}}_q(a_1\mathcal{X}\left(\vartheta \right)+a_2\mathcal{Y}\left(\vartheta \right))=a_1{\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right)+a_2{\mathfrak{D}}_q\mathcal{Y}\left(\vartheta \right),\hfill \\ \left(\mathrm{iii}\right)& {\mathfrak{D}}_q\left(\mathcal{X}\mathcal{Y}\right)\left(\vartheta \right)=\mathcal{X}\left(q\vartheta \right){\mathfrak{D}}_q\mathcal{Y}\left(\vartheta \right)+\mathcal{Y}\left(\vartheta \right){\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right)=\mathcal{X}\left(\vartheta \right){\mathfrak{D}}_q\mathcal{Y}\left(\vartheta \right)+\mathcal{Y}\left(q\vartheta \right){\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right),\hfill \\ \left(\mathrm{iv}\right)& {\mathfrak{D}}_q\left({\displaystyle \frac{\mathcal{X}\left(\vartheta \right)}{\mathcal{Y}\left(\vartheta \right)}}\right)={\displaystyle \frac{\mathcal{Y}\left(\vartheta \right){\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right)-\mathcal{X}\left(\vartheta \right){\mathfrak{D}}_q\mathcal{Y}\left(\vartheta \right)}{\mathcal{Y}\left(\vartheta \right)\mathcal{Y}\left(q\vartheta \right)}}={\displaystyle \frac{\mathcal{Y}\left(q\vartheta \right){\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right)-\mathcal{X}\left(q\vartheta \right){\mathfrak{D}}_q\mathcal{Y}\left(\vartheta \right)}{\mathcal{Y}\left(\vartheta \right)\mathcal{Y}\left(q\vartheta \right)}}.\hfill \end{array}$$

The functions

$$e_q\left(\vartheta \right)=\sum _{\omega =0}^{\infty }{\displaystyle \frac{{\vartheta }^{\omega }}{{\left[\omega \right]}_q!}}{,0<\left|q\right|<1,\left|\vartheta \right|<|1-q|}^{-1},$$

(6)

$$E_q\left(\vartheta \right)=\sum _{\omega =0}^{\infty }{q}^{\left({\displaystyle \genfrac{}{}{0pt}{}{\omega }{2}}\right)}{\displaystyle \frac{{\vartheta }^{\omega }}{{\left[\omega \right]}_q!}},0<\left|q\right|<1,\vartheta \in \mathbb{C}$$

(7)

are termed q-exponential functions [13] and possess the following identities:

$$\begin{array}{c}\hfill {\mathfrak{D}}_q{e}_q\left(\vartheta \right)=e_q\left(\vartheta \right),{\mathfrak{D}}_q{E}_q\left(\vartheta \right)=E_q\left(q\vartheta \right),\end{array}$$

(8)

$$\begin{array}{c}\hfill e_q\left(\vartheta \right)E_q(-\vartheta )=E_q\left(\vartheta \right)e_q(-\vartheta )=1,\end{array}$$

(9)

$$\begin{array}{c}\hfill {\mathfrak{D}}_q^{\rho }{e}_q\left(\lambda \vartheta \right)={\lambda }^{\rho }{e}_q\left(\lambda \vartheta \right),\rho \ge 1.\end{array}$$

(10)

The q-definite integral of a function $\mathcal{X}\left(\vartheta \right)$ can be expressed as follows [6]:

$${\int }_0^{\zeta }\mathcal{X}\left(\vartheta \right)d_q\vartheta =(1-q)\zeta \sum _{\omega =0}^{\infty }{\displaystyle \frac{1}{q^{\omega +1}}}\mathcal{X}\left({\displaystyle \frac{\zeta }{q^{\omega +1}}}\right),0<q<1,\zeta \in \mathbb{R}$$

(11)

and

$${\int }_{\eta }^{\zeta }{\mathfrak{D}}_q\mathcal{X}\left(\vartheta \right)d_q\vartheta =\mathcal{X}\left(\zeta \right)-\mathcal{X}\left(\eta \right).$$

(12)

The q-dilation operator [3] is defined for any function $\mathcal{X}\left(\vartheta \right)$ as

$${\mathfrak{T}}_{\vartheta }^{\varrho }\mathcal{X}\left(\vartheta \right)=\mathcal{X}\left(q^{\varrho }\vartheta \right),\vartheta \in \mathbb{C},\varrho \in \mathbb{R},0<q<1$$

(13)

and fulfills the following property:

$${\mathfrak{T}}_{\vartheta }^{-1}{\mathfrak{T}}_{\vartheta }^1\mathcal{X}\left(\vartheta \right)=\mathcal{X}\left(\vartheta \right).$$

(14)

The q-derivative of $e_q\left(\vartheta \varkappa \right)$ is given by

$${\mathfrak{D}}_q{e}_q\left(\vartheta {\varkappa }^r\right)=\vartheta {\varkappa }^{r-1}{\mathfrak{T}}_{(\vartheta ;r)}{e}_q\left(\vartheta {\varkappa }^r\right),$$

where

$${\mathfrak{T}}_{(\vartheta ;r)}={\displaystyle \frac{1-q^r{\mathfrak{T}}_{\vartheta }^r}{1-q{\mathfrak{T}}_{\vartheta }}}=1+q{\mathfrak{T}}_{\vartheta }+\dots +q^r{\mathfrak{T}}_{\vartheta }^r.$$

Bell polynomials are widely regarded as highly significant among special polynomials, owing to their broad applications in diverse mathematical contexts (see [14,15,16,17,18]). Furthermore, they hold critical importance in the analysis of water wave phenomena, driving progress in fields such as mechanical engineering, energy development, marine and offshore engineering, hydraulic engineering, and other related disciplines [19,20,21,22]. The Bell polynomials $Be{l}_{\omega }\left(\vartheta \right)$ are defined by [17,18]

$$e^{\vartheta (e^{\varkappa }-1)}=\sum _{\omega =0}^{\infty }Be{l}_{\omega }\left(\vartheta \right){\displaystyle \frac{{\varkappa }^{\omega }}{\omega !}}.$$

(15)

The 2-variable Bell polynomials (2VBelP) $Be{l}_{\omega }({\vartheta }_1,{\vartheta }_2)$ are defined by [23]

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

(16)

The q-analogue of the Bell polynomials $Be{l}_{\omega }\left(\vartheta \right)$ (denoted $Be{l}_{\omega ,q}\left(\vartheta \right)$) [24] has the following generating function:

$$e_q\left(\vartheta (e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }Be{l}_{\omega ,q}\left(\vartheta \right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(17)

For $\vartheta =1$, the q-Bell polynomials (q-BelP) $Be{l}_{\omega ,q}\left(\vartheta \right)$ reduce to the q-Bell numbers $Be{l}_{\omega ,q}$ [25], that is,

$$e_q\left((e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }Be{l}_{\omega ,q}{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(18)

Further, the 2-variable q-BelP polynomials (2Vq-BelP) $Be{l}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ are defined as

$$e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_2(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }Be{l}_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(19)

Bell numbers, polynomials, and their q-analogues have been extensively studied by numerous researchers, with significant properties explored in works such as [16,17,18,23,25] and references therein. Additionally, these mathematical objects play a vital role in diverse fields, including analytic number theory, physics, and related disciplines.

The 2-variable general polynomials (2VGP) ${\mathbb{G}}_{\omega }({\vartheta }_1,{\vartheta }_2)$ are specified by [26]:

$$e^{{\vartheta }_1\varkappa }\psi ({\vartheta }_2,\varkappa )=\sum _{\omega =0}^{\infty }{\mathbb{G}}_{\omega }({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{\omega !}},{\mathbb{G}}_0({\vartheta }_1,{\vartheta }_2)=1,$$

(20)

where

$$\psi ({\vartheta }_2,\varkappa )=\sum _{\omega =0}^{\infty }{\psi }_{\omega }\left({\vartheta }_2\right){\displaystyle \frac{{\varkappa }^{\omega }}{\omega !}},{\psi }_0\left({\vartheta }_2\right)\ne 0.$$

(21)

The q-analogue of the 2VGP ${\mathbb{G}}_{\omega }({\vartheta }_1,{\vartheta }_2)$, is denoted by ${\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ [27] and defined by

$$e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )=\sum _{\omega =0}^{\infty }{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},{\mathbb{G}}_{0,q}({\vartheta }_1,{\vartheta }_2)=1,$$

(22)

where

$${\psi }_q({\vartheta }_2,\varkappa )=\sum _{\omega =0}^{\infty }{\psi }_{\omega ,q}\left({\vartheta }_2\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},{\psi }_{0,q}\left({\vartheta }_2\right)\ne 0.$$

(23)

Significant members of the 2-variable q-general polynomials (2Vq-GP) family are obtained by making suitable selections for the function ${\psi }_q({\vartheta }_2,\varkappa )$, such as

• If ${\psi }_q({\vartheta }_2,\varkappa )=e_q\left({\vartheta }_2{\varkappa }^r\right)$, the 2Vq-GP ${\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ reduce to the q-Gould–Hopper polynomials (q-GH) ${\mathcal{H}}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2)$ [27], where ${\mathcal{H}}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2)$ are defined by

  $$e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^r\right)=\sum _{\omega =0}^{\infty }{\mathcal{H}}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}$$

  (24)

  and series representation as

  $${\mathcal{H}}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2)={\left[\omega \right]}_q\sum _{\varrho =0}^{\left[{\displaystyle \frac{\omega }{r}}\right]}{\displaystyle \frac{{\vartheta }_2^{\varrho }{\vartheta }_1^{\omega -r\varrho }}{{\left[\varrho \right]}_q!{[\omega -r\varrho ]}_q!}}.$$

  (25)
  For $r=2$, the q-GHP ${\mathcal{H}}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2)$ reduce to 2-variable q-Hermite polynomials (2vq-HP) [28].
• If ${\psi }_q({\vartheta }_2,\varkappa )=C_{0,q}\left({\vartheta }_2\varkappa \right)$, the 2Vq-GP ${\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ reduce to 2-variable q-Laguerre polynomials (2vq-LP) ${\mathtt{L}}_{\omega ,q}({\vartheta }_2,{\vartheta }_1)$ [29], where ${\mathtt{L}}_{\omega ,q}({\vartheta }_2,{\vartheta }_1)$ are defined by

  $$e_q\left({\vartheta }_1\varkappa \right)C_{0,q}\left({\vartheta }_2\varkappa \right)=\sum _{\omega =0}^{\infty }{\mathtt{L}}_{\omega ,q}({\vartheta }_2,{\vartheta }_1){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},$$

  (26)

  where $C_{0,q}\left({\vartheta }_2\varkappa \right)$ denotes the 0th order q-Tricomi functions, which are defined by

  $$C_{0,q}\left(\vartheta \right)=\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{(-1)}^{\varrho }{\vartheta }^{\varrho }}{{({\left[\varrho \right]}_q!)}^2}}.$$

  (27)
• If ${\psi }_q({\vartheta }_2,\varkappa )=A_q\left(\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^2\right)$, the 2Vq-GP ${\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ reduce to the q-Hermite–Appell polynomials (q-HAP) ${}_{\mathcal{H}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ [30], where ${}_{\mathcal{H}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ are defined by

  $$A_q\left(\varkappa \right)e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^2\right)=\sum _{\omega =0}^{\infty }{}_{\mathcal{H}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

  (28)
• If ${\psi }_q({\vartheta }_2,\varkappa )={\displaystyle \frac{1}{1-{\vartheta }_2{\varkappa }^s}}$, the 2Vq-GP ${\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ reduce to 2-variable q-truncated exponential polynomials of order s (2Vq-TEP) $e_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2)$ [31], where ${}_{\mathcal{H}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ are defined by

  $${\displaystyle \frac{e_q\left({\vartheta }_1\varkappa \right)}{1-{\vartheta }_2{\varkappa }^s}}=\sum _{\omega =0}^{\infty }{e}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

  (29)
• If ${\psi }_q({\vartheta }_2,\varkappa )={\displaystyle \frac{A_q\left(\varkappa \right)}{1-{\vartheta }_2{\varkappa }^s}}$, the 2Vq-GP ${\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ reduce to 2-variable q-truncated exponential Appell polynomials (2Vq-TEAP) ${}_{e^{\left(s\right)}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ [31], where ${}_{e^{\left(s\right)}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ are defined by

  $${\displaystyle \frac{A_q\left(\varkappa \right)}{1-{\vartheta }_2{\varkappa }^s}}{e}_q\left({\vartheta }_1\varkappa \right)=\sum _{\omega =0}^{\infty }{}_{e^{\left(s\right)}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

  (30)
  For $s=2$, the 2Vq-TEAP ${}_{e^{\left(s\right)}}A_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ reduce to q-truncated exponential Appell polynomials (q-TEAP) [28], which are defined by

  $${\displaystyle \frac{A_q\left(\varkappa \right)}{1-{\varkappa }^s}}{e}_q\left({\vartheta }_1\varkappa \right)=\sum _{\omega =0}^{\infty }{}_{e^{\left(s\right)}}A_{\omega ,q}\left({\vartheta }_1\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

  (31)

The family of q-special polynomials, including q-Bell polynomials, q-general polynomials, and specific members of the q-general class such as q-Gould–Hopper, q-Laguerre, q-Hermite, and q-truncated exponential polynomials form a dynamically growing area of mathematical research. This expansion is supported by extensive investigations, as highlighted in works such as Alam et al. [27] introduced the 2-variable q-General–Appell polynomials and studied the related properties; Raza et al. [28] established the 2-variable q-Hermite polynomials; Fadel et al. [29] investigated the Bivariate q-Laguerre–Appell polynomials; Costabile et al. [31] defined and studied the q-truncated exponential Appell polynomials, and Wani et al. [32] discussed the certain advancement in multidimensional q-Hermite polynomials.

In recent years, researchers have increasingly focused on developing and analyzing considerable classes of q-special polynomials, particularly generalized classes, as evidenced by works such as [27,30,31,33,34] and related references.

The key novelty of this work lies in the establishment of a new generalized family of q-special polynomials. This versatile family can be considered a generalization of many special polynomials, including the q-Bell, 2-variable q-general, q-Gould–Hopper, q-Laguerre, q-Hermite, q-Hermite–Appell, q-truncated exponential, q-Gould–Hopper–Bell, q-Laguerre–Bell, q-Hermite–Bell, q-Hermite–Appell–Bell, and q-truncated exponential-Bell polynomials, which expands the theory of special function by introducing and systematically studying a new, broad class of q-special polynomials and their properties.

This article is structured as follows: In Section 2, we employ the replacement technique to construct the class of q-general Bell polynomials by synthesizing standard polynomials, q-general polynomials, and q-Bell polynomials. The generating function is introduced. Additionally, explicit series formulations linked to these polynomials are rigorously established. In Section 3, the associated recurrence relations, quasi-monomial characteristics, and a q-differential equation are systematically analyzed and formalized. Section 4 explores specific applications, including the q-Gould–Hopper–Bell polynomials and the q-truncated exponential-Bell polynomials, with analogous theoretical frameworks developed for each case. Furthermore, computational tools in Mathematica are utilized to investigate the zero distributions of these polynomials, accompanied by graphical visualizations that provide numerical and illustrative perspectives on their properties. For a consolidated overview of the polynomials and their associated symbols, refer to Table A1.

2. q-General Bell Polynomials

Assume that $0<q<1$. This section presents a new class of generalized q-special polynomials, the q-general Bell polynomials (q-GBelP), characterized by their generating functions. Certain series representations are also formulated.

Theorem 1.

The q-general Bell polynomials ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ have the following generating function:

$$e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},0<q<1.$$

(32)

Proof. 

We begin with the generating relation (19) and replace ${\vartheta }_2$ with ${\vartheta }_3$. Subsequently, we expand the function $e_q\left({\vartheta }_1\varkappa \right)$ on the l.h.s. of the resulting equation and replace the powers of ${\vartheta }_1$, i.e., ${\vartheta }_1^0,{\vartheta }_1,{\vartheta }_1^2,\dots ,{\vartheta }_1^{\omega }$ with the corresponding polynomials ${\mathbb{G}}_{0,q}({\vartheta }_1,{\vartheta }_2),{\mathbb{G}}_{1,q}({\vartheta }_1,{\vartheta }_2),{\mathbb{G}}_{2,q}({\vartheta }_1,{\vartheta }_2),\dots ,{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)$ in the l.h.s. and ${\vartheta }_1$ by ${\mathbb{G}}_{1,q}({\vartheta }_1,{\vartheta }_2)$ in the r.h.s. We then have

$$\begin{array}{cc}\hfill (1+{\mathbb{G}}_{1,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{\varkappa }{{\left[1\right]}_q!}}+{\mathbb{G}}_{2,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^2}{{\left[2\right]}_q!}}+\dots & +{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}+\dots )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\hfill \\ & =\sum _{\omega =0}^{\infty }Be{l}_{\omega ,q}({\mathbb{G}}_{1,q}({\vartheta }_1,{\vartheta }_2),{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(33)

Further, by summing the series on the l.h.s. and then substituting Equation (22) into the resulting equation, we obtain

$$e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }Be{l}_{\omega ,q}({\mathbb{G}}_{1,q}({\vartheta }_1,{\vartheta }_2),{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(34)

Finally, indicating the resultant q-GBelP as ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, that is,

$$Be{l}_{\omega ,q}({\mathbb{G}}_{1,q}({\vartheta }_1,{\vartheta }_2),{\vartheta }_3)={}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),$$

(35)

the assertion in Equation (32) is proven. □

Remark 1.

Setting ${\vartheta }_1=0$ in generating relation (32), we obtain the 2-variable q-general Bell polynomials ${}_{\mathbb{G}}\mathcal{B}\mathit{el}_{\omega ,q}({\vartheta }_2,{\vartheta }_3)$, as defined by

$${\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathcal{B}\mathit{el}_{\omega ,q}({\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(36)

By suitable selections of variables and the function ${\psi }_q({\vartheta }_2,\varkappa )$, we obtain certain members of the q-general Bell polynomials. These members are mentioned in Table 1.

Table 1. Some members of the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$.

q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\mathit{\omega },\mathit{q}}({\mathit{\vartheta }}_1,{\mathit{\vartheta }}_2,{\mathit{\vartheta }}_3)$	Choice	Well-Known Polynomials	Generating Function
$e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)$	${\vartheta }_3=0$	2-variable q-general	$e^{{\vartheta }_1\varkappa }\psi ({\vartheta }_2,\varkappa )$
$={\displaystyle \sum _{\omega =0}^{\infty }}{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}$		polynomials	$={\displaystyle \sum _{\omega =0}^{\infty }}{\mathbb{G}}_{\omega }({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{\omega !}}$.
	${\psi }_q({\vartheta }_2,\varkappa )=1$	2-variable q-Bell	$e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)$
		polynomials	$={\displaystyle \sum _{\omega =0}^{\infty }}Be{l}_{\omega ,q}({\vartheta }_1,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$
	${\psi }_q({\vartheta }_2,\varkappa )=e_q\left({\vartheta }_2{\varkappa }^r\right)$	q-Gould–Hopper	$e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^r\right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)$
		-Bell polynomials	$={\displaystyle \sum _{\omega =0}^{\infty }}{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$
	${\psi }_q({\vartheta }_2,\varkappa )=C_{0,q}\left({\vartheta }_2\varkappa \right)$	q-Laguerre–Bell	$e_q\left({\vartheta }_1\varkappa \right)C_{0,q}\left({\vartheta }_2\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)$
		polynomials	$={\displaystyle \sum _{\omega =0}^{\infty }}{}_{\mathtt{L}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$
	${\psi }_q({\vartheta }_2,\varkappa )=A_q\left(\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^2\right)$	q-Hermite–Appell	$A_q\left(\varkappa \right)e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^2\right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)$
		-Bell polynomials	$={\displaystyle \sum _{\omega =0}^{\infty }}{}_{\mathcal{H}A}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$
	${\psi }_q({\vartheta }_2,\varkappa )={\displaystyle \frac{1}{1-{\vartheta }_2{\varkappa }^s}}$	2-variable q-truncated	${\displaystyle \frac{e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)}{1-{\vartheta }_2{\varkappa }^s}}$
		exponential Bell polynomials	$={\displaystyle \sum _{\omega =0}^{\infty }}{}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$
	${\psi }_q({\vartheta }_2,\varkappa )={\displaystyle \frac{A_q\left(\varkappa \right)}{1-{\vartheta }_2{\varkappa }^s}}$	2-variable q-truncated exponential	${\displaystyle \frac{A_q\left(\varkappa \right)e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)}{1-{\vartheta }_2{\varkappa }^s}}$
		Appell–Bell polynomials	$={\displaystyle \sum _{\omega =0}^{\infty }}{}_{eA}\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$

Next, in view of generating function (32), we establish certain series representations for the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$.

Theorem 2.

For $\omega \ge 0$ and $0\le \varrho \le \omega$, the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ satisfy the following series representation:

$${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\mathbb{G}}_{\omega -\varrho ,q}({\vartheta }_1,{\vartheta }_2)Be{l}_{\varrho ,q}\left({\vartheta }_3\right),0<q<1.$$

(37)

Proof. 

Based on Equations (17) and (22), Equation (32) can be expressed as

$$\sum _{\omega =0}^{\infty }{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\sum _{\varrho =0}^{\infty }Be{l}_{\varrho ,q}\left({\vartheta }_3\right){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}=\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},$$

(38)

which, when employing the Cauchy product rule [35], leads to

$$\sum _{\omega =0}^{\infty }\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2)Be{l}_{\varrho ,q}\left({\vartheta }_3\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}=\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(39)

Comparing coefficients of like powers of $\varkappa$ in the above equation establishes assertion (37). □

Similarly, the following theorem can be proven.

Theorem 3.

For $\omega \ge 0$ and $0\le \varrho \le \omega$, the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ satisfy the following series representations:

$$\begin{array}{cc}\hfill & {}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\psi }_{\omega -\varrho ,q}\left({\vartheta }_2\right)Be{l}_{\varrho ,q}({\vartheta }_1,{\vartheta }_3),\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & {}_{\mathbb{G}}\mathcal{B}\mathit{el}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_2,{\vartheta }_3){\vartheta }_1^{\varrho }.\hfill \end{array}$$

(41)

Theorem 4.

For $\omega \ge 0$, $0\le \varrho \le \omega$, and $0\le \alpha \le \varrho$, the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ satisfy the following series representation:

$${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{\vartheta }_1^{\omega -\varrho }{\psi }_{\varrho -\alpha ,q}\left({\vartheta }_2\right)Be{l}_{\alpha ,q}\left({\vartheta }_3\right).$$

(42)

Proof. 

In view of Equations (6), (17), (23), (32) and Cauchy product rule, we have

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}& =\left(\sum _{\omega =0}^{\infty }{\vartheta }_1^{\omega }{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }{\psi }_{\varrho ,q}\left({\vartheta }_2\right){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\left(\sum _{\alpha =0}^{\infty }Be{l}_{\alpha ,q}\left({\vartheta }_3\right){\displaystyle \frac{{\varkappa }^{\alpha }}{{\left[\alpha \right]}_q!}}\right)\hfill \\ & =\left(\sum _{\omega =0}^{\infty }{\vartheta }_1^{\omega }{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{\psi }_{\varrho -\alpha ,q}\left({\vartheta }_2\right)Be{l}_{\alpha ,q}\left({\vartheta }_3\right){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\hfill \\ & =\sum _{\omega =0}^{\infty }\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{\vartheta }_1^{\omega -\varrho }{\psi }_{\varrho -\alpha ,q}\left({\vartheta }_2\right)Be{l}_{\alpha ,q}\left({\vartheta }_3\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(43)

Comparing coefficients of like powers of $\varkappa$ in the above equation establishes assertion (42). □

The q-analogue of Stirling numbers of the second kind is given by [24]

$${\displaystyle \frac{{(e_q\left(\varkappa \right)-1)}^{\varrho }}{{\left[\varrho \right]}_q!}}=\sum _{\omega =\varrho }^{\infty }{\mathbb{S}}_{2,q}(\omega ,\varrho ){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(44)

Theorem 5.

For $\omega \ge 0$, $0\le \varrho \le \omega$, and $0\le \alpha \le \varrho$, we have

$${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\vartheta }_3^{\alpha }{\mathbb{G}}_{\omega -\varrho ,q}({\vartheta }_1,{\vartheta }_2){\mathbb{S}}_{2,q}(\varrho ,\alpha ).$$

(45)

Proof. 

In view of Equations (6), (22), (32), (44), and the Cauchy product rule, we have

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}& =e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\hfill \\ & =\left(\sum _{\omega =0}^{\infty }{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\alpha =0}^{\infty }{(e_q\left(\varkappa \right)-1)}^{\alpha }{\displaystyle \frac{{\vartheta }_3^{\alpha }}{{\left[\alpha \right]}_q!}}\right)\hfill \\ & =\left(\sum _{\omega =0}^{\infty }{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\alpha =0}^{\infty }{\vartheta }_3^{\alpha }\sum _{\varrho =\alpha }^{\infty }{\mathbb{S}}_{2,q}(\varrho ,\alpha ){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\hfill \\ & =\left(\sum _{\omega =0}^{\infty }{\mathbb{G}}_{\omega ,q}({\vartheta }_1,{\vartheta }_2){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }\sum _{\alpha =0}^{\varrho }{\vartheta }_3^{\alpha }{\mathbb{S}}_{2,q}(\varrho ,\alpha ){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\hfill \\ & =\sum _{\omega =0}^{\infty }\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\mathbb{G}}_{\omega -\varrho ,q}({\vartheta }_1,{\vartheta }_2){\vartheta }_3^{\alpha }{\mathbb{S}}_{2,q}(\varrho ,\alpha ){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(46)

Comparison of the coefficients of identical powers of $\varkappa$ in the above equation establishes the assertion in Equation (45). □

Now, let us recall the 2-variable q-tangent polynomials ${\mathcal{C}}_{\varrho ,\sigma ,q}\left({\vartheta }_1\right)$ [36], which are given by

$$\left({\displaystyle \frac{2}{e_q\left(\sigma \varkappa \right)+1}}\right)e_q\left({\vartheta }_1\varkappa \right)=\sum _{\varrho =0}^{\infty }{\mathcal{C}}_{\varrho ,\sigma ,q}\left({\vartheta }_1\right){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}},\left|\alpha \varkappa \right|<\pi ,\alpha \in {\mathbb{R}}^{+}.$$

(47)

Theorem 6.

The q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ satisfy the following representation:

$$\begin{array}{cc}\hfill & {}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\hfill \\ & ={\displaystyle \frac{1}{2}}\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q\left[\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_2,{\vartheta }_3){\sigma }^{\alpha }{\mathcal{C}}_{\varrho -\alpha ,\sigma ,q}\left({\vartheta }_1\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}+{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\mathcal{C}}_{\varrho ,\sigma ,q}\right].\hfill \end{array}$$

(48)

Proof. 

Based on Equations (6), (36) and (47), Equation (32) can be expressed as

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}& ({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}=e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\hfill \\ & ={\displaystyle \frac{e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )}{e_q\left(\sigma \varkappa \right)+1}}\left(e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\right)(e_q\left(\sigma \varkappa \right)+1)\hfill \\ & ={\displaystyle \frac{1}{2}}\left[\left(\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{\sigma }^{\alpha }{\mathcal{C}}_{\varrho -\alpha ,\sigma ,q}\left({\vartheta }_1\right){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\right.\hfill \\ & \left.+\left(\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\alpha =0}^{\infty }{\mathcal{C}}_{\varrho ,\sigma ,q}{\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\right]\hfill \\ & ={\displaystyle \frac{1}{2}}\sum _{\omega =0}^{\infty }\left[\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_2,{\vartheta }_3){\sigma }^{\alpha }{\mathcal{C}}_{\varrho -\alpha ,\sigma ,q}\left({\vartheta }_1\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right.\hfill \\ & \left.+\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\mathcal{C}}_{\varrho ,\sigma ,q}\right]{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(49)

From the above equation, comparing the coefficients of ϰ results in Equation (48). □

3. Recurrence Formulae and Quasi-Monomial Characteristics

In this section, we establish some q-recurrence relations and quasi-monomial properties associated with q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$.

Theorem 7.

The q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ admit the following q-recurrence formula:

$${\mathfrak{D}}_{q,{\vartheta }_1}^{\rho }\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\displaystyle \frac{{\left[\omega \right]}_q!}{{[\omega -\rho ]}_q!}}{}_{\mathbb{G}}\mathit{Bel}_{\omega -\rho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3).$$

(50)

Proof. 

By taking the ${\rho }^{th}$ q-partial derivative of generating relation (32) with respect to ${\vartheta }_1$ and utilizing Equation (10), we have

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{\mathfrak{D}}_{q,{\vartheta }_1}^{\rho }\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}& ={\varkappa }^{\rho }\left\{e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\right\}\hfill \\ & =\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega +\rho }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(51)

The simplification of Equation (51) followed by a comparison of the coefficients of ϰ on both sides establishes the result (50). □

Remark 2.

Taking $\rho =1$ in (50), we have

$${\mathfrak{D}}_{q,{\vartheta }_1}\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\left[\omega \right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\rho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3).$$

(52)

Theorem 8.

For q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, the following q-recurrence formula holds:

$${\mathfrak{D}}_{q,{\vartheta }_2}\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\xi }_{\varrho ,q}{}_{\mathbb{G}}\mathit{Bel}_{\omega -\rho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),$$

(53)

where

$${\displaystyle \frac{{\mathfrak{D}}_{q,{\vartheta }_2}{\psi }_q({\vartheta }_2,\varkappa )}{{\psi }_q({\vartheta }_2,\varkappa )}}=\sum _{\omega =0}^{\infty }{\xi }_{\omega ,q}{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(54)

Proof. 

Consider the generating function

$$\begin{array}{c}\hfill {\mathsf{G}}_q({\vartheta }_1,{\vartheta }_2,{\vartheta }_3,\varkappa )=e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\end{array}$$

(55)

Applying the q-partial derivative ${\mathfrak{D}}_{q,{\vartheta }_2}$ to generating relation (55), it follows that:

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{q,{\vartheta }_2}\left\{{\mathsf{G}}_q({\vartheta }_1,{\vartheta }_2,{\vartheta }_3,\varkappa )\right\}& =e_q\left({\vartheta }_1\varkappa \right)\left({\mathfrak{D}}_{q,{\vartheta }_2}{\psi }_q({\vartheta }_2,\varkappa )\right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\hfill \\ & ={\mathsf{G}}_q({\vartheta }_1,{\vartheta }_2,{\vartheta }_3,\varkappa )\left({\displaystyle \frac{{\mathfrak{D}}_{q,{\vartheta }_2}{\psi }_q({\vartheta }_2,\varkappa )}{{\psi }_q({\vartheta }_2,\varkappa )}}\right).\hfill \end{array}$$

(56)

In view of the assumption in Equation (54), Equation (56) can be expressed as

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{\mathfrak{D}}_{q,{\vartheta }_2}\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}& =\left(\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }{\xi }_{\varrho ,q}{\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\hfill \\ & =\sum _{\omega =0}^{\infty }\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\xi }_{\varrho ,q}{}_{\mathbb{G}}\mathit{Bel}_{\omega -\rho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(57)

Comparing the coefficients of ϰ in (57) leads to the statement (53). □

Theorem 9.

The q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ admit the following q-recurrence formula:

$${\mathfrak{D}}_{q,{\vartheta }_3}\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\left[\omega \right]}_q\sum _{\varrho =0}^{\omega }{\displaystyle \frac{1}{{[\varrho +1]}_q}}{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega -1}{\varrho }}\right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho -1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3).$$

(58)

Proof. 

By taking the q-partial derivative of generating relation (32) with respect to ${\vartheta }_3$ and utilizing Equation (10), we find

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{\mathfrak{D}}_{q,{\vartheta }_3}& \left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}=(e_q\left(\varkappa \right)-1)\left\{e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\right\}\hfill \\ & =\left(\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\varkappa }^{\varrho +1}}{{[\varrho +1]}_q!}}\right)\hfill \\ & =\sum _{\omega =0}^{\infty }\sum _{\varrho =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega +\varrho +1}}{{[\varrho +1]}_q!{\left[\omega \right]}_q!}}\hfill \\ & =\sum _{\omega =0}^{\infty }\sum _{\varrho =0}^{\omega }{\displaystyle \frac{{\left[\omega \right]}_q}{{[\varrho +1]}_q}}{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega -1}{\varrho }}\right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho -1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(59)

Comparing the coefficients of ϰ in (59) leads to the statement (58). □

Similarly, we can prove the following result.

Theorem 10.

The q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ admit the following q-recurrence formula:

$${\mathfrak{D}}_{q,{\vartheta }_3}\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}+{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{}_{\mathbb{G}}\mathit{Bel}_{\omega -\rho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3).$$

(60)

Theorem 11.

The q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ satisfy the following q-recurrence relation:

$$\begin{array}{cc}\hfill & {}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\hfill \\ & =\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q\left(Be{l}_{\omega -\varrho ,q}(q{\vartheta }_1,{\vartheta }_3){\psi }_{\varrho +1,q}\left({\vartheta }_2\right)+{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_2,{\vartheta }_3){\vartheta }_1^{\varrho +1}+{\vartheta }_3{q}^{\omega -\varrho }{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right).\hfill \end{array}$$

(61)

Proof. 

Consider the generating function

$${\mathsf{G}}_q({\vartheta }_1,{\vartheta }_2,{\vartheta }_3,\varkappa )=e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(62)

The q-partial differentiation of Equation (62) with respect to ϰ gives

$$\begin{array}{c}\hfill {\mathfrak{D}}_{q,\varkappa }\left({\mathsf{G}}_q({\vartheta }_1,{\vartheta }_2,{\vartheta }_3,\varkappa )\right)=\left[e_q\left(q{\vartheta }_1\varkappa \right)\left({\mathfrak{D}}_{q,\varkappa }{\psi }_q({\vartheta }_2,\varkappa )\right)+{\vartheta }_1{\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_1\varkappa \right)\right]e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\\ \hfill +{\vartheta }_3{e}_q\left(q{\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,q\varkappa )e_q\left(\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right).\end{array}$$

(63)

Proceeding, we take the q-partial derivative with respect to ϰ on both sides of Equation (23), which leads to

$$D_{q,\varkappa }{\psi }_q({\vartheta }_2,\varkappa )=\sum _{\omega =0}^{\infty }{\psi }_{\omega +1,q}\left({\vartheta }_2\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

(64)

In view of Equations (6), (19), (36), (62) and (64), Equation (63) can be expressed as

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}& =\left(\sum _{\omega =0}^{\infty }Be{l}_{\omega ,q}(q{\vartheta }_1,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }{\psi }_{\varrho +1,q}\left({\vartheta }_2\right){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\hfill \\ & +\left(\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }{\vartheta }_1^{\varrho +1}{\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right)\hfill \\ & +{\vartheta }_3\left(\sum _{\omega =0}^{\infty }{q}^{\omega }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right)\left(\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}\right),\hfill \end{array}$$

(65)

which (using the Cauchy product rule) yields

$$\begin{array}{cc}\hfill & \sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\hfill \\ & =\sum _{\omega =0}^{\infty }\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q\left(Be{l}_{\omega -\varrho ,q}(q{\vartheta }_1,{\vartheta }_3){\psi }_{\varrho +1,q}\left({\vartheta }_2\right)+{}_{\mathbb{G}}\mathcal{B}\mathit{el}_{\omega -\varrho ,q}({\vartheta }_2,{\vartheta }_3){\vartheta }_1^{\varrho +1}+{\vartheta }_3{q}^{\omega -\varrho }{}_{\mathbb{G}}\mathit{Bel}_{\omega -\varrho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right){\displaystyle \frac{{\varkappa }^{\varrho }}{{\left[\varrho \right]}_q!}}.\hfill \end{array}$$

(66)

Finally, comparing coefficients of ϰ results in Equation (61). □

Theorem 12.

For q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, the following q-integral formula holds:

$${\int }_{\eta }^{\zeta }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)d_q{\vartheta }_1={\displaystyle \frac{{}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}(\zeta ,{\vartheta }_2,{\vartheta }_3)-{}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}(\eta ,{\vartheta }_2,{\vartheta }_3)}{{[\omega +1]}_q}}.$$

(67)

Proof. 

Taking the q-definite integration of generating relation (32) with respect to ${\vartheta }_1$ and using Equation (10), we obtain

$$\begin{array}{cc}\hfill \sum _{\omega =0}^{\infty }{\int }_{\eta }^{\zeta }& {}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)d_q{\vartheta }_1{\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\hfill \\ & ={\displaystyle \frac{1}{\varkappa }}\left(\left\{e_q\left(\zeta \varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\right\}-\left\{e_q\left(\eta \varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\right\}\right)\hfill \\ & =\sum _{\omega =0}^{\infty }\left({}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}(\zeta ,{\vartheta }_2,{\vartheta }_3)-{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}(\eta ,{\vartheta }_2,{\vartheta }_3)\right){\displaystyle \frac{{\varkappa }^{\omega -1}}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(68)

Simplifying and comparing the coefficients of ϰ on both sides of Equation (68) results in Equation (67). □

In line with the monomiality principle [37], a polynomial set ${\psi }_{\omega ,q}\left(\vartheta \right)$$(\omega \in \mathbb{N},\vartheta \in \mathbb{C})$ is termed quasi-monomial if it is possible to define “q-multiplicative” (${\widehat{M}}_q$) and “q-derivative” (${\widehat{P}}_q$) operators for which

$${\widehat{M}}_q\left\{{\psi }_{\omega ,q}\left(\vartheta \right)\right\}={\psi }_{\omega +1,q}\left(\vartheta \right),$$

(69)

$${\widehat{P}}_q\left\{{\psi }_{\omega ,q}\left(\vartheta \right)\right\}={\left[\omega \right]}_q{\psi }_{\omega -1,q}\left(\vartheta \right),$$

(70)

respectively, for all $\omega \in \mathbb{N}$. Moreover, these q-operators satisfy the following relations

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

$${\widehat{M}}_q{\widehat{P}}_q\left\{{\psi }_{\omega ,q}\left(\vartheta \right)\right\}={\left[\omega \right]}_q{\psi }_{\omega ,q}\left(\vartheta \right),$$

(71)

$${\widehat{P}}_q{\widehat{M}}_q\left\{{\psi }_{\omega ,q}\left(\vartheta \right)\right\}={[\omega +1]}_q{\psi }_{\omega ,q}\left(\vartheta \right),$$

(72)

$$exp\left(\varkappa {\widehat{M}}_q\right)\left\{1\right\}=\sum _{\omega =0}^{\infty }{\psi }_{\omega ,q}\left(\vartheta \right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},\left|\varkappa \right|<\infty .$$

(73)

The principle of monomiality, being a powerful tool for the analysis of q-special polynomials, motivates our examination of the quasi-monomiality of the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$.

Theorem 13.

For q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, the associated q-multiplicative and q-derivative operators demonstrating their quasi-monomial nature are

$${\widehat{M}}_{qGBelP}={\vartheta }_1+{\displaystyle \frac{{\psi }_q^{\prime }({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{\mathfrak{T}}_{{\vartheta }_1}+{\vartheta }_3{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\displaystyle \frac{{\psi }_q({\vartheta }_2,q{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{\mathfrak{T}}_{{\vartheta }_1}$$

(74)

and

$${\widehat{P}}_{qGBelP}={\mathfrak{D}}_{q,{\vartheta }_1},$$

(75)

respectively.

Proof. 

Obviously, we have

$${\mathfrak{D}}_{q,{\vartheta }_1}\left[e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\right]=\varkappa \left[e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\right].$$

(76)

The q-partial derivative of Equation (32) with respect to ϰ yields

$$\begin{array}{c}[e_q\left(q{\vartheta }_1\varkappa \right)\left({\mathfrak{D}}_{q,\varkappa }{\psi }_q({\vartheta }_2,\varkappa )\right)+{\vartheta }_1{\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_1\varkappa \right)]e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\hfill \\ +{\vartheta }_3{e}_q\left(q{\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,q\varkappa )e_q\left(\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},\hfill \end{array}$$

(77)

which on using (13) becomes

$$\begin{array}{cc}\hfill \left({\vartheta }_1+{\displaystyle \frac{{\psi }_q^{\prime }({\vartheta }_2,\varkappa )}{{\psi }_q({\vartheta }_2,\varkappa )}}{\mathfrak{T}}_{{\vartheta }_1}+{\vartheta }_3{\displaystyle \frac{{\psi }_q({\vartheta }_2,q\varkappa )}{{\psi }_q({\vartheta }_2,\varkappa )}}{e}_q\left(\varkappa \right){\mathfrak{T}}_{{\vartheta }_1}\right)& e_q\left({\vartheta }_1\varkappa \right){\psi }_q({\vartheta }_2,\varkappa )e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)\hfill \\ & =\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},\hfill \end{array}$$

(78)

which on using relation (76) on the left-hand side becomes

$$\begin{array}{cc}\hfill ({\vartheta }_1+{\displaystyle \frac{{\psi }_q^{\prime }({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{\mathfrak{T}}_{{\vartheta }_1}+& {\vartheta }_3{\displaystyle \frac{{\psi }_q({\vartheta }_2,q{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\mathfrak{T}}_{{\vartheta }_1})\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\hfill \\ & =\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(79)

Matching the coefficients of corresponding powers of ϰ in Equation (79), we obtain

$$\begin{array}{cc}\hfill \left({\vartheta }_1+{\displaystyle \frac{{\psi }_q^{\prime }({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{\mathfrak{T}}_{{\vartheta }_1}+{\vartheta }_3{\displaystyle \frac{{\psi }_q({\vartheta }_2,q{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\mathfrak{T}}_{{\vartheta }_1}\right)& {}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\hfill \\ & ={}_{\mathbb{G}}\mathit{Bel}_{\omega +1,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \end{array}$$

(80)

Using Equations (69) (for q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$) in (80) we obtain assertion (74).

Using relation (32) in expression (76) gives

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{q,{\vartheta }_1}\left[\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right]& =\varkappa \left[\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}\right]\hfill \\ & =\sum _{\omega =0}^{\infty }{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega +1}}{{\left[\omega \right]}_q!}}.\hfill \end{array}$$

(81)

Next, after replacing ω with $\omega -1$ on the right-hand side, we equate the coefficients of corresponding powers of ϰ (on both sides) to obtain

$${\mathfrak{D}}_{q,{\vartheta }_1}\left\{{}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}=\varkappa {}_{\mathbb{G}}\mathit{Bel}_{\omega -\rho ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3).$$

(82)

Using Equations (70) (for q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$) in (82), we obtain assertion (75). □

Theorem 14.

The q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ satisfy the following differential equation:

$$\left({\vartheta }_1{\mathfrak{D}}_{q,{\vartheta }_1}+{\displaystyle \frac{{\psi }_q^{\prime }({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{\mathfrak{D}}_{q,{\vartheta }_1}{\mathfrak{T}}_{{\vartheta }_1}+{\vartheta }_3{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\displaystyle \frac{{\psi }_q({\vartheta }_2,q{\mathfrak{D}}_{q,{\vartheta }_1})}{{\psi }_q({\vartheta }_2,{\mathfrak{D}}_{q,{\vartheta }_1})}}{\mathfrak{D}}_{q,{\vartheta }_1}{\mathfrak{T}}_{{\vartheta }_1}-\omega \right){}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0.$$

(83)

Proof. 

Based on Equation (71) (for ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$), utilizing operators (74) and (75) yields the asserted result (83). □

4. Applications

This section presents applications associated with the our family, q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$. Specific illustrative examples are analyzed to demonstrate their properties and utility. Additionally, the zero distributions of the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ are investigated, offering insights into their structural and analytical behavior.

Examples

The q-Gould–Hopper, q-Laguerre, q-Hermite, q-Appell, and q-truncated exponential polynomials are vital components in diverse expansions and approximation formulas, forming the foundation for advancements in classical and numerical analysis. These polynomials, along with their associated q-numbers, are further instrumental in the analytic theory of numbers. Specific members of the broader q-general family can be derived through a suitable selection of the function ${\psi }_q({\vartheta }_2,\varkappa )$. Here, we present key results for the corresponding members of the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, by examining illustrative examples that highlight their structural and analytic properties.

Example 1.

q-Gould–Hopper–Bell Polynomials.

Taking ${\psi }_q({\vartheta }_2,\varkappa )=e_q\left({\vartheta }_2{\varkappa }^r\right)$ in Equation (32), we obtain

$$e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^r\right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},$$

(84)

where ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ represents the q-Gould–Hopper–Bell polynomials (q-GHBelP).

In view of Equations (37), (40), (41), (42), (45) and (48), certain series representations of the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ are given as follows:

$$\begin{array}{cc}\hfill & {}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\mathcal{H}}_{\omega -\varrho ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2)Be{l}_{\varrho ,q}\left({\vartheta }_3\right),\hfill \end{array}$$

(85)

$$\begin{array}{cc}& {}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\left[\omega \right]}_q!\sum _{\varrho =0}^{\left[{\displaystyle \frac{\omega }{r}}\right]}{\displaystyle \frac{{\vartheta }_2^{\varrho }Be{l}_{\omega -r\varrho ,q}({\vartheta }_1,{\vartheta }_3)}{{\left[\varrho \right]}_q!{[\omega -r\varrho ]}_q!}},\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill & {}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{}_{\mathcal{H}}\mathcal{B}\mathit{el}_{\omega -\varrho ,q}^{\left(r\right)}({\vartheta }_2,{\vartheta }_3){\vartheta }_1^{\varrho },\hfill \end{array}$$

(87)

$$\begin{array}{cc}& {}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\left[{\displaystyle \frac{\varrho }{r}}\right]}{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\displaystyle \frac{{\left[\varrho \right]}_q!{\vartheta }_1^{\omega -\varrho }{\vartheta }_2^{\alpha }Be{l}_{\varrho -r\alpha ,q}\left({\vartheta }_3\right)}{{\left[\alpha \right]}_q!{[\varrho -r\alpha ]}_q!}},\hfill \end{array}$$

(88)

$$\begin{array}{cc}& {}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\vartheta }_3^{\alpha }{\mathcal{H}}_{\omega -\varrho ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2){\mathbb{S}}_{2,q}(\varrho ,\alpha ),\hfill \end{array}$$

(89)

$$\begin{array}{c}{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\\ ={\displaystyle \frac{1}{2}}\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q\left[\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{}_{\mathcal{H}}\mathit{Bel}_{\omega -\varrho ,q}^{\left(r\right)}({\vartheta }_2,{\vartheta }_3){\sigma }^{\alpha }{\mathcal{C}}_{\varrho -\alpha ,\sigma ,q}\left({\vartheta }_1\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}+{}_{\mathcal{H}}\mathit{Bel}_{\omega -\varrho ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\mathcal{C}}_{\varrho ,\sigma ,q}\right].\hfill \end{array}$$

(90)

From Equations (50), (53), (58), (60), (61), and (67), we obtain the following q-recurrence relations for the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$:

$$\begin{array}{cc}\hfill & {\mathfrak{D}}_{q,{\vartheta }_1}^{\rho }\left\{{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\displaystyle \frac{{\left[\omega \right]}_q!}{{[\omega -\rho ]}_q!}}{}_{\mathcal{H}}\mathit{Bel}_{\omega -\rho ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \\ & {\mathfrak{D}}_{q,{\vartheta }_2}\left\{{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\displaystyle \frac{{\left[\omega \right]}_q!}{{[\omega -r]}_q!}}{}_{\mathcal{H}}\mathit{Bel}_{\omega -r,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \\ & {\mathfrak{D}}_{q,{\vartheta }_3}\left\{{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\left[\omega \right]}_q\sum _{\varrho =0}^{\omega }{\displaystyle \frac{1}{{[\varrho +1]}_q}}{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega -1}{\varrho }}\right]}_q{}_{\mathcal{H}}\mathit{Bel}_{\omega -\varrho -1,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \\ & {\mathfrak{D}}_{q,{\vartheta }_3}\left\{{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}+{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{}_{\mathcal{H}}\mathit{Bel}_{\omega -\varrho -1,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \\ & {}_{\mathcal{H}}\mathit{Bel}_{\omega +1,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\left[\omega \right]}_q!\sum _{\varrho =0}^{\left[{\displaystyle \frac{\omega +1}{r}}\right]}{\displaystyle \frac{{\vartheta }_2^{\varrho }{\left[r\varrho \right]}_{q}Be{l}_{\omega -r\varrho +1,q}(q{\vartheta }_1,{\vartheta }_3)}{{\left[\varrho \right]}_q!{[\omega -r\varrho +1]}_q!}}\hfill \\ & +\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q\left({}_{\mathcal{H}}\mathcal{B}\mathit{el}_{\omega -\varrho ,q}^{\left(r\right)}({\vartheta }_2,{\vartheta }_3){\vartheta }_1^{\varrho +1}+{\vartheta }_3{q}^{\omega -\varrho }{}_{\mathcal{H}}\mathcal{B}\mathit{el}_{\omega -\varrho ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right),\hfill \\ & {\int }_{\eta }^{\zeta }{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)d_q{\vartheta }_1={\displaystyle \frac{{}_{\mathcal{H}}\mathit{Bel}_{\omega +1,q}^{\left(r\right)}(\zeta ,{\vartheta }_2,{\vartheta }_3)-{}_{\mathcal{H}}\mathit{Bel}_{\omega +1,q}^{\left(r\right)}(\eta ,{\vartheta }_2,{\vartheta }_3)}{{[\omega +1]}_q}}.\hfill \end{array}$$

For q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, the associated q-multiplicative and q-derivative operators demonstrating their quasi-monomial nature are

$${\widehat{M}}_{qGHBelP}={\vartheta }_1+{\vartheta }_2{\mathfrak{D}}_{q,{\vartheta }_1}^{r-1}{\mathfrak{T}}_{{(\vartheta }_1;r)}{\mathfrak{T}}_{{\vartheta }_1}+{\vartheta }_3{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\mathfrak{T}}_{{\vartheta }_2}^r{\mathfrak{T}}_{{\vartheta }_1}$$

and

$${\widehat{P}}_{qGHBelP}={\mathfrak{D}}_{q,{\vartheta }_1},$$

respectively.

From Equation (83), we find the following differential equation, which is satisfied by he q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$:

$$\left({\vartheta }_1{\mathfrak{D}}_{q,{\vartheta }_1}+{\vartheta }_2{\mathfrak{D}}_{q,{\vartheta }_1}^r{\mathfrak{T}}_{{(\vartheta }_1;r)}{\mathfrak{T}}_{{\vartheta }_1}+{\vartheta }_3{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\mathfrak{D}}_{q,{\vartheta }_1}{\mathfrak{T}}_{{\vartheta }_2}^r{\mathfrak{T}}_{{\vartheta }_1}-\omega \right){}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0.$$

In view of (84), we list the first five members of the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ for $r=2$ as

$$\begin{array}{cc}\hfill & {}_{\mathcal{H}}\mathit{Bel}_{0,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=1,\hfill \\ & {}_{\mathcal{H}}\mathit{Bel}_{1,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\vartheta }_1+{\vartheta }_3,\hfill \\ & {}_{\mathcal{H}}\mathit{Bel}_{2,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=(q+1){\vartheta }_1{\vartheta }_3+q{\vartheta }_2+{\vartheta }_1^2+{\vartheta }_2+{\vartheta }_3^2+{\vartheta }_3,\hfill \\ & {}_{\mathcal{H}}\mathit{Bel}_{3,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\left(q^2+q+1\right){\vartheta }_3\left(q{\vartheta }_2+{\vartheta }_1^2+{\vartheta }_1+{\vartheta }_2\right)+(q+1)\left(q^2+q+1\right){\vartheta }_1{\vartheta }_2\hfill \\ & +{\displaystyle \frac{\left(q^2+q+1\right){\vartheta }_3^2(q{\vartheta }_1+{\vartheta }_1+2)}{q+1}}+{\vartheta }_1^3+{\vartheta }_3^3+{\vartheta }_3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {}_{\mathcal{H}}\mathit{Bel}_{4,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\displaystyle \frac{1}{1+q}}\left\{(1+q){\vartheta }_1^4+{(1+q)}^2\left(1+q+2q^2+q^3+q^4\right){\vartheta }_2^2\right.\hfill \\ & +{(1+q)}^2\left(1+q^2\right){\vartheta }_1^3{\vartheta }_3+{(1+q)}^2\left(1+q+2q^2+q^3+q^4\right){\vartheta }_2{\vartheta }_3(1+{\vartheta }_3)\hfill \\ & +\left(1+2q+3q^2+3q^3+2q^4+q^5\right){\vartheta }_1^2\left({\vartheta }_2+q{\vartheta }_2+{\vartheta }_3+{\vartheta }_3^2\right)\hfill \\ & +{\vartheta }_3\left(q^4{\vartheta }_3+3q^3{\vartheta }_3(1+{\vartheta }_3)+{(1+{\vartheta }_3)}^3+q{(1+{\vartheta }_3)}^3+q^2{\vartheta }_3(4+3{\vartheta }_3)\right)\hfill \\ & \left.+\left(1+q+q^2+q^3\right){\vartheta }_1{\vartheta }_3\left({\vartheta }_2+3q^3{\vartheta }_2+q^4{\vartheta }_2+{(1+{\vartheta }_3)}^2+2q^2(2{\vartheta }_2+{\vartheta }_3)+q\left(3{\vartheta }_2+{(1+{\vartheta }_3)}^2\right)\right)\right\}.\hfill \end{array}$$

Next, we explore the distributions of zeros and present graphical illustrations of the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ for specific parameter values and indices.

Certain interesting zeros of the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$, for $r=2$ and $\omega =60$ are shown in Figure 1.

Figure 1. Zeros of ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$ with degree $\omega =60$.

The stacking structures of approximation zeros for the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$ ($r=2$, $1\le \omega \le 40$) yield 3D structures, as shown in Figure 2.

Figure 2. Stacking structure zeros of ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$. This figure shows the 3D plot of the zeros of the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$, for $r=2$ and $1\le \omega \le 40$.

Example 2.

q-Truncated Exponential-Bell Polynomials.

Taking ${\psi }_q({\vartheta }_2,\varkappa )={\displaystyle \frac{1}{1-{\vartheta }_2{\varkappa }^s}}$ in Equation (32), we obtain

$${\displaystyle \frac{e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)}{1-{\vartheta }_2{\varkappa }^s}}=\sum _{\omega =0}^{\infty }{}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}},$$

where ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ are referred to as the q-truncated exponential Bell polynomials of order s (q-TEBelP).

In view of Equations (37), (40), (41), (42), (45) and (48), certain series representations of the q-TEBelP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ are given as follows:

$$\begin{array}{cc}\hfill & {}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{e}_{\omega -\varrho ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2)Be{l}_{\varrho ,q}\left({\vartheta }_3\right),\hfill \\ & {}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\left[\omega \right]}_q!\sum _{\varrho =0}^{\left[{\displaystyle \frac{\omega }{s}}\right]}{\displaystyle \frac{{\vartheta }_2^{\varrho }Be{l}_{\omega -s\varrho ,q}({\vartheta }_1,{\vartheta }_3)}{{[\omega -s\varrho ]}_q!}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \\ \hfill & {}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{}_e\mathcal{B}\mathit{el}_{\omega -\varrho ,q}^{\left(s\right)}({\vartheta }_2,{\vartheta }_3){\vartheta }_1^{\varrho },\hfill \\ & {}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\left[{\displaystyle \frac{\varrho }{s}}\right]}{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\displaystyle \frac{{\left[\varrho \right]}_q!{\vartheta }_1^{\omega -\varrho }{\vartheta }_2^{\alpha }Be{l}_{\varrho -s\alpha ,q}\left({\vartheta }_3\right)}{{[\varrho -s\alpha ]}_q!}},\hfill \\ & {}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{\vartheta }_3^{\alpha }{e}_{\omega -\varrho ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2){\mathbb{S}}_{2,q}(\varrho ,\alpha ),\hfill \\ \hfill & {}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\hfill \\ & ={\displaystyle \frac{1}{2}}\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q\left[\sum _{\alpha =0}^{\varrho }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\varrho }{\alpha }}\right]}_q{}_e\mathcal{B}\mathit{el}_{\omega -\varrho ,q}^{\left(s\right)}({\vartheta }_2,{\vartheta }_3){\sigma }^{\alpha }{\mathcal{C}}_{\varrho -\alpha ,\sigma ,q}\left({\vartheta }_1\right){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}+{}_e\mathit{Bel}_{\omega -\varrho ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\mathcal{C}}_{\varrho ,\sigma ,q}\right].\hfill \end{array}$$

From Equations (50), (53), (58), (60), (61) and (67), we obtain the following q-recurrence relations for the q-TEBelP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$:

$$\begin{array}{cc}\hfill & {\mathfrak{D}}_{q,{\vartheta }_1}^{\rho }\left\{{}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\displaystyle \frac{{\left[\omega \right]}_q!}{{[\omega -\rho ]}_q!}}{}_e\mathit{Bel}_{\omega -\rho ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \\ & {\mathfrak{D}}_{q,{\vartheta }_3}\left\{{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}={\left[\omega \right]}_q\sum _{\varrho =0}^{\omega }{\displaystyle \frac{1}{{[\varrho +1]}_q}}{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega -1}{\varrho }}\right]}_q{}_{\mathcal{H}}\mathit{Bel}_{\omega -\varrho -1,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \\ & {\mathfrak{D}}_{q,{\vartheta }_3}\left\{{}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right\}+{}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q{}_e\mathit{Bel}_{\omega -\varrho ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3),\hfill \\ & {}_e\mathit{Bel}_{\omega +1,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\left[\omega \right]}_q!\sum _{\varrho =0}^{\left[{\displaystyle \frac{\omega +1}{s}}\right]}{\displaystyle \frac{{\vartheta }_2^{\varrho }{\left[s\varrho \right]}_{q}Be{l}_{\omega -s\varrho +1,q}(q{\vartheta }_1,{\vartheta }_3)}{{[\omega -s\varrho +1]}_q!}}\hfill \\ & +\sum _{\varrho =0}^{\omega }{\left[{\displaystyle \genfrac{}{}{0pt}{}{\omega }{\varrho }}\right]}_q\left({}_e{\mathcal{B}}^{\left(s\right)}\mathit{el}_{\omega -\varrho ,q}({\vartheta }_2,{\vartheta }_3){\vartheta }_1^{\varrho +1}+{\vartheta }_3{q}^{\omega -\varrho }{}_e\mathit{Bel}_{\omega -\varrho ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right),\hfill \\ & {\int }_{\eta }^{\zeta }{}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)d_q{\vartheta }_1={\displaystyle \frac{{}_e\mathit{Bel}_{\omega +1,q}^{\left(s\right)}(\zeta ,{\vartheta }_2,{\vartheta }_3)-{}_e\mathit{Bel}_{\omega +1,q}^{\left(s\right)}(\eta ,{\vartheta }_2,{\vartheta }_3)}{{[\omega +1]}_q}}.\hfill \end{array}$$

For q-TEBelP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, the associated q-multiplicative and q-derivative operators demonstrating their quasi-monomial nature are

$${\widehat{M}}_{qTEBelP}={\vartheta }_1+{\displaystyle \frac{{\left[s\right]}_q{\vartheta }_2{\mathfrak{D}}_{q,{\vartheta }_1}^{s-1}}{1-{\vartheta }_2{q}^s{\mathfrak{D}}_{q,{\vartheta }_1}^s}}+{\vartheta }_3{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\displaystyle \frac{{\left[s\right]}_q{\vartheta }_2{\mathfrak{D}}_{q,{\vartheta }_1}^s}{1-{\vartheta }_2{q}^s{\mathfrak{D}}_{q,{\vartheta }_1}^s}}{\mathfrak{T}}_{{\vartheta }_1}$$

and

$${\widehat{P}}_{qTEBelP}={\mathfrak{D}}_{q,{\vartheta }_1},$$

respectively.

From Equation (83), we find the following differential equation, which is satisfied by the q-TEBelP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$:

$$\left({\vartheta }_1{\mathfrak{D}}_{q,{\vartheta }_1}+{\displaystyle \frac{{\left[s\right]}_q{\vartheta }_2{\mathfrak{D}}_{q,{\vartheta }_1}^s}{1-{\vartheta }_2{q}^s{\mathfrak{D}}_{q,{\vartheta }_1}^s}}+{\vartheta }_3{e}_q\left({\mathfrak{D}}_{q,{\vartheta }_1}\right){\displaystyle \frac{{\left[s\right]}_q{\vartheta }_2{\mathfrak{D}}_{q,{\vartheta }_1}^{s+1}}{1-{\vartheta }_2{q}^s{\mathfrak{D}}_{q,{\vartheta }_1}^s}}{\mathfrak{T}}_{{\vartheta }_1}-\omega \right){}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0.$$

In view of (84), we list the first five members of the q-TEBelP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ for $s=3$ as

$$\begin{array}{cc}\hfill & {}_e\mathit{Bel}_{0,q}^{\left(3\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=1,\hfill \\ & {}_e\mathit{Bel}_{1,q}^{\left(3\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\vartheta }_1+{\vartheta }_3,\hfill \\ & {}_e\mathit{Bel}_{2,q}^{\left(3\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=(q+1){\vartheta }_1{\vartheta }_3+{\vartheta }_1^2+{\vartheta }_3({\vartheta }_3+1),\hfill \\ & {}_e\mathit{Bel}_{3,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\displaystyle \frac{\left(1+q+q^2\right)(2+{\vartheta }_1+q{\vartheta }_1){\vartheta }_3^2}{1+q}}+{\vartheta }_1^3+(1+q)\left(1+q+q^2\right){\vartheta }_2\hfill \\ & +{\vartheta }_3+\left(1+q+q^2\right){\vartheta }_1(1+{\vartheta }_1){\vartheta }_3+{\vartheta }_3^3,\hfill \\ & {}_e\mathit{Bel}_{4,q}^{\left(3\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)={\vartheta }_1^4+(1+q)\left(1+q^2\right){\vartheta }_1^3{\vartheta }_3+\left(1+q^2\right)\left(1+q+q^2\right){\vartheta }_1^2{\vartheta }_3(1+{\vartheta }_3)\hfill \\ & +{\displaystyle \frac{{\vartheta }_3\left(1+q+{(1+q)}^3\left(1+q^2\right)\left(1+q+q^2\right){\vartheta }_2+\left(1+q^2\right)(3+q(3+q)){\vartheta }_3+3(1+q)\left(1+q^2\right){\vartheta }_3^2+(1+q){\vartheta }_3^3\right)}{1+q}}\hfill \\ & +\left(1+q^2\right){\vartheta }_1\left({(1+q)}^2\left(1+q+q^2\right){\vartheta }_2+{\vartheta }_3\left(1+q+2\left(1+q+q^2\right){\vartheta }_3+(1+q){\vartheta }_3^2\right)\right).\hfill \end{array}$$

Next, we explore the distributions of zeros and present graphical illustrations of the q-GHBelP ${}_{\mathcal{H}}\mathit{Bel}_{\omega ,q}^{\left(r\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ for specific parameter values and indices.

Certain interesting zeros of the q-TEBelP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$ for $s=3$ and $\omega =60$ are shown in Figure 3.

Figure 3. Zeros of ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$ with degree $\omega =60$.

The stacking structures of approximation zeros of the q-TEBelP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$, for $s=3$ and $1\le \omega \le 40$ give 3D structures, which are presented in Figure 4.

Figure 4. Stacking structure zeros of ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$. This figure shows the 3D plot of the zeros of the TGHBelATP ${}_e\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)=0$, for $s=3$ and $1\le \omega \le 40$.

Further, we can present more examples for our established family q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ based on suitable selections for the function ${\psi }_q({\vartheta }_2,\varkappa )$, as follows:

• If ${\psi }_q({\vartheta }_2,\varkappa )=C_{0,q}\left({\vartheta }_2\varkappa \right)$, the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ reduce to q-Laguerre–Bell polynomials (q-LBelP) ${}_{\mathtt{L}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, which are defined by

  $$e_q\left({\vartheta }_1\varkappa \right)C_{0,q}\left({\vartheta }_2\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathtt{L}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$
• If ${\psi }_q({\vartheta }_2,\varkappa )=A_q\left(\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^2\right)$, the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ reduce to the q-Hermite–Appell–Bell polynomials (q-HABelP) ${}_{\mathcal{H}A}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, which are defined by

  $$A_q\left(\varkappa \right)e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_2{\varkappa }^2\right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)=\sum _{\omega =0}^{\infty }{}_{\mathcal{H}A}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$
• If ${\psi }_q({\vartheta }_2,\varkappa )={\displaystyle \frac{A_q\left(\varkappa \right)}{1-{\vartheta }_2{\varkappa }^s}}$, the q-GBelP ${}_{\mathbb{G}}\mathit{Bel}_{\omega ,q}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ reduce to q-truncated exponential-Appell–Bell polynomials of order s (q-TEABelP) ${}_{eA}\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, which are defined by

  $${\displaystyle \frac{A_q\left(\varkappa \right)e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)}{1-{\vartheta }_2{\varkappa }^s}}=\sum _{\omega =0}^{\infty }{}_{eA}\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$
  For $s=2$, the q-TEABelP ${}_{eA}\mathit{Bel}_{\omega ,q}^{\left(s\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ reduce to q-truncated exponential-Appell–Bell polynomials of order 2 (q-TEABelP) ${}_{eA}\mathit{Bel}_{\omega ,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$, which are defined by

  $${\displaystyle \frac{A_q\left(\varkappa \right)e_q\left({\vartheta }_1\varkappa \right)e_q\left({\vartheta }_3(e_q\left(\varkappa \right)-1)\right)}{1-{\vartheta }_2{\varkappa }^2}}=\sum _{\omega =0}^{\infty }{}_{eA}\mathit{Bel}_{\omega ,q}^{\left(2\right)}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3){\displaystyle \frac{{\varkappa }^{\omega }}{{\left[\omega \right]}_q!}}.$$

5. Conclusions

The generalized formulation of special polynomials has attracted considerable interest within the research community. This paper introduces a new class of special polynomials, the q-general Bell polynomials, and investigates their properties. The key findings are as follows:

• Introduction of a New Polynomial Family: A generalized family of q-special polynomials, termed q-general Bell polynomials, were introduced through generating function and series representations.
• Fundamental Properties: Some explicit representations and q-recurrence relations for these polynomials were established, which helps to clarify their combinatorial nature.
• Structural Characteristics: The quasi-monomial properties of the q-general Bell polynomials were examined, and the differential equation that governs them was constructed.
• Specific Instances and Adaptability: The versatility of this new family of polynomials was demonstrated through the analysis of specific cases, including the q-Gould–Hopper–Bell and q-truncated exponential-Bell polynomials. For each of these specific instances, analogous structural and combinatorial results were derived.
• Applications: This study demonstrates the properties and applications of the q-general Bell polynomials through the analysis of specific examples. Furthermore, an investigation into their zero distributions provides key insights into their structural and analytical characteristics.
• Computational Analysis: To provide numerical and visual insights into the behavior of these polynomials, computational analyses were performed using Mathematica to explore the distribution of their zeros and to generate graphical representations.

In summary, this study provides a unifying theoretical framework for a diverse set of special polynomial families. It also lays the groundwork for future investigations into the theoretical extensions and practical applications of these polynomials in fields such as mathematical physics, combinatorics, and approximation theory. Potential future research could relate our introduced family to the results that are given in [38,39,40]. Further studies may also explore the degenerate forms of these polynomials.