Finding the q-Appell Convolution of Certain Polynomials Within the Context of Quantum Calculus

Abstract

This article introduces the theory of three-variable q-truncated exponential Gould–Hopper-based Appell polynomials by employing a generating function approach that incorporates q-calculus functions. This study further explores these polynomials by using a computational algebraic approach. The determinant form, recurrences, and differential equations are proven. Relationships with the monomiality principle are given. Finally, graphical representations are presented to illustrate the behavior and potential applications of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials.

1. Introduction

Special polynomials are fundamental mathematical constructs that serve as versatile tools across various disciplines. Their significance lies in their capacity to simplify complex mathematical problems and provide elegant solutions to real-world challenges. These polynomials are particularly effective in approximating intricate functions, rendering them invaluable in numerical analysis and computational mathematics. Their orthogonality properties facilitate the efficient representation of functions in series expansions, which is crucial in areas such as signal processing and data compression. Furthermore, special polynomials frequently emerge as solutions to differential equations that model physical phenomena, thereby bridging the gap between abstract mathematics and practical applications in physics and engineering. The utility of special polynomials extends beyond their computational advantages. Their associated generating functions offer a compact means of representing infinite sequences, thereby aiding the study of combinatorial problems and probability distributions. The recurrence relations governing these polynomials not only allow for efficient computation but also reveal profound mathematical structures and patterns. Additionally, the symmetry properties exhibited by many special polynomials provide insights into the underlying mathematical and physical principles of the systems that they describe. This combination of theoretical elegance and practical utility renders special polynomials indispensable in fields ranging from quantum mechanics to financial modeling, cementing their status as essential tools in modern mathematics and its applications. Gould–Hopper polynomials are the subject of extensive research due to their distinctive characteristics and applicability in fields such as approximation theory, differential equations, and quantum mechanics. Gould–Hopper polynomials, which represent a generalization of Hermite polynomials, have been the focus of comprehensive investigations regarding their mathematical properties and practical applications.

Recent advancements in special polynomial, q-Appell, and truncated multivariate polynomial families have underscored their utility in operator theory, approximation, and discrete modeling [1,2,3,4,5]. These investigations have provided a comprehensive framework within which the proposed generalization can be contextualized. The 2-variable truncated exponential polynomials (2VTPs) denoted by $e_n^{\left(r\right)}(\tau ,\varphi )$ are specified by the generating relation [6,7,8]

$${\displaystyle \frac{e^{\tau t}}{1-\varphi t^r}}=\sum _{n=0}^{\infty }{e}_n^{\left(r\right)}(\tau ,\varphi ){\displaystyle \frac{t^n}{n!}},$$

(1)

where

$$e_n^{\left(r\right)}(\tau ,\varphi )=n!\sum _{k=0}^{\left[{\displaystyle \frac{n}{r}}\right]}{\displaystyle \frac{{\varphi }^k{\tau }^{n-rk}}{(n-rk)!}}.$$

(2)

Very recently, in [9], Özat et al. presented the general form of truncated exponential-based general-Appell polynomials, utilizing the framework of two-variable general-Appell polynomials as follows:

$$A\left(t\right)e^{\tau t}\psi \left(\varphi ,t\right){\displaystyle \frac{1}{1-\gamma t^m}}=\sum _{s=0}^{\infty }{}_{e^{\left(m\right)}}{T}_s(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{n!}}.$$

In addition, the authors successfully derived explicit and determinant representations, along with operators for lowering and raising, a recurrence relation, a differential equation, and various summation formulas pertinent to these polynomials.

The field of quantum calculus, alternatively known as q-calculus, represents an area of burgeoning research interest. For $q\to 1^{-},$ quantum calculus converges to classical calculus. In contemporary scientific discourse, quantum calculus has exhibited its utility across diverse disciplines, encompassing mathematical sciences, quantum physics, quantum mechanics, quantum algebra, approximation theory, and operator theory, among other domains. We now provide some basic properties of quantum analysis. For the following properties and more details, please see [1,10,11,12,13]. Assume that $0<q<1$. The q-shifted factorial ${(\gamma ;q)}_v$ is defined as

$${(\vartheta ;q)}_u=\prod _{w=1}^{u-1}(1-q^w\vartheta ),(u\in \mathbb{N}),{(\vartheta ;q)}_0=1.$$

(3)

The q-analogue of a number $\gamma$ is given by

$${\left[\vartheta \right]}_q={\displaystyle \frac{1-q^{\vartheta }}{1-q}},0<q<1;\vartheta \in \mathbb{C}.$$

(4)

The q-factorial is given by

$${\left[\vartheta \right]}_q!=\left\{\begin{array}{ccc}\prod _{w=1}^{\vartheta }{\left[w\right]}_q,\vartheta \ge 1& & \\ & & \\ 1,& \gamma =0& \end{array}\right..$$

(5)

The following equality holds:

$${(\tau \pm \gamma )}_q^v=\sum _{s=0}^v{\left({\displaystyle \genfrac{}{}{0pt}{}{v}{s}}\right)}_q{q}^{\left({\displaystyle \genfrac{}{}{0pt}{}{v-s}{2}}\right)}{\tau }^s{(\pm \gamma )}^{v-s},$$

(6)

where

$${\left({\displaystyle \genfrac{}{}{0pt}{}{\vartheta }{w}}\right)}_q={\displaystyle \frac{{\left[\vartheta \right]}_q!}{{\left[\vartheta -w\right]}_q!{\left[w\right]}_q!}}.$$

The following equalities hold:

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

(7)

and

$$E_q\left(\tau \right)=\sum _{w=0}^{\infty }{\displaystyle \frac{q^{\left(\genfrac{}{}{0pt}{}{w}{2}\right)}{\tau }^w}{{\left[j\right]}_q!}},0<q<1.$$

(8)

The following equality holds:

$$e_q\left(\tau \right)E_q\left(\varphi \right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(\tau +\varphi )}_q^n}{{\left[n\right]}_q!}}.$$

(9)

Hence,

$$e_q\left(\tau \right)E_q(-\tau )=1.$$

(10)

The following equality holds:

$${\widehat{D}}_{q,\tau }h\left(\tau \right)={\displaystyle \frac{h\left(q\tau \right)-h\left(\tau \right)}{\tau \left(1-q\right)}},0<q<1,\tau \ne 0.$$

(11)

In particular, we have

$${\widehat{D}}_{q,\tau }{\tau }^w={\left[w\right]}_q{\tau }^{w-1},$$

(12)

$${\widehat{D}}_{q,\tau }{e}_q\left(\alpha \tau \right)=\alpha e_q\left(\alpha \tau \right),\alpha \in \mathbb{C},$$

(13)

and

$${\widehat{D}}_{q,\tau }^w{e}_q\left(\alpha \tau \right)={\alpha }^w{e}_q\left(\alpha \tau \right),w\in \mathbb{N},\alpha \in \mathbb{C},$$

(14)

where ${\widehat{D}}_{q,\tau }^w$ denotes the $w^{th}$-order q-derivative with respect to $\tau$.

The q-derivative concerning the product of the functions $f\left(\tau \right)$ and $g\left(\tau \right)$ has been thoroughly examined in the existing literature.

$${\widehat{D}}_{q,\tau }\left(h\left(\tau \right)k\left(\tau \right)\right)=h\left(\tau \right){\widehat{D}}_{q,\tau }k\left(\tau \right)+k\left(q\tau \right){\widehat{D}}_{q,\tau }h\left(\tau \right)$$

(15)

and

$${\widehat{D}}_{q,\tau }\left({\displaystyle \frac{h\left(\tau \right)}{k\left(\tau \right)}}\right)={\displaystyle \frac{k\left(q\tau \right){\widehat{D}}_{q,\tau }h\left(\tau \right)-h\left(q\tau \right){\widehat{D}}_{q,\tau }k\left(\tau \right)}{k\left(\tau \right)k\left(q\tau \right)}}.$$

(16)

The q-integral of a function $h\left(\tau \right)$ is introduced as

$${\int }_0^{a}h\left(\tau \right)d_q\tau =(1-q)a\sum _{w=0}^{\infty }{q}^{w}h\left(a{q}^w\right).$$

(17)

By virtue of (17), we get

$${\int }_0^{\tau }{\lambda }^w{d}_q\lambda =(1-q)\tau \sum _{n=0}^{\infty }{\tau }^w{q}^{n(w+1)}$$

$$={\displaystyle \frac{{\tau }^{w+1}(1-q)}{1-q^{w+1}}}={\displaystyle \frac{{\tau }^{w+1}}{{[w+1]}_q}},w\in \mathbb{N}\bigsqcup \left\{0\right\}.$$

(18)

In particular, ${\widehat{D}}_{q,\tau }^{-1}\left\{1\right\}=\tau$, and then

$${\left({\widehat{D}}_{q,\tau }^{-1}\right)}^r\left\{1\right\}={\displaystyle \frac{{\tau }^r}{{\left[r\right]}_q!}},r\in \mathbb{N}\bigsqcup \left\{0\right\}.$$

(19)

The q-GHPs, symbolized as $H_{n,q}^{\left(m\right)}(\tau ,\varphi )$, are established through the use of the generating function [14]

$$e_q\left(\tau t\right)e_q\left(\varphi t^m\right)=\sum _{n=0}^{\infty }{H}_{n,q}^{\left(m\right)}(\tau ,\varphi ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}},$$

(20)

and the series definition is

$$H_{n,q}^{\left(m\right)}(\tau ,\varphi )={\left[n\right]}_q!\sum _{k=0}^{\left[{\displaystyle \frac{n}{m}}\right]}{\displaystyle \frac{{\varphi }^k{\tau }^{n-mk}}{{\left[k\right]}_q!{[n-mk]}_q!}}.$$

(21)

For $H_{n,q}^{\left(m\right)}(\tau ,\varphi )$, the following equality holds (see [15]):

$$H_{n,q}^{\left(m\right)}(\tau ,\varphi )=e_q\left(\varphi D_{q,\tau }^m\right)\left\{{\tau }^n\right\}.$$

(22)

The q-dilatation operator $T_z$ is characterized by its action on any function of the complex variable z as follows [2,16]:

$$T_z^{k}f\left(z\right)=f\left(q^{k}z\right),k\in \mathbb{R},$$

(23)

satisfying the property

$$T_z^{-1}{T}_z^{1}f\left(z\right)=f\left(z\right).$$

(24)

The following equality holds [15]:

$${\widehat{D}}_{q,t}{e}_q\left(\varphi t^m\right)=\varphi t^{m-1}{T}_{(\varphi ;m)}{e}_q\left(\varphi t^m\right).$$

(25)

where

$$T_{(\varphi ;m)}={\displaystyle \frac{1-q^m{T}_{\varphi }^m}{1-q{T}_{\varphi }}}=1+q{T}_{\varphi }+\cdots +q^{m-1}{T}_{\varphi }^{m-1}.$$

Al Salam provides a comprehensive explanation of the generating function for the q-Appell polynomials ${\mathcal{A}}_{n,q}\left(\tau \right)$, as articulated in the formula found in [17,18].

$${\mathcal{A}}_q\left(t\right)e_q\left(\tau t\right)=\sum _{w=0}^{\infty }{\mathcal{A}}_{w,q}\left(\tau \right){\displaystyle \frac{t^w}{{\left[w\right]}_q!}},$$

(26)

$${\mathcal{A}}_q\left(t\right)=\sum _{w=0}^{\infty }{\mathcal{A}}_{w,q}{\displaystyle \frac{t^w}{{\left[w\right]}_q!}},{\mathcal{A}}_q\left(t\right)\ne 0,A_{0,q}=1.$$

(27)

The following represents a suitable option for specific individuals within the category of q-Appell polynomials (Table 1).

Table 1. Certain q-APF.

S. No.	q-APF	GF	${\mathcal{A}}_{\mathit{q}}\left(\mathit{t}\right)$
I.	The q-BP [3,19]	${\displaystyle \frac{\delta }{e_q\left(\delta \right)-1}{e}_q\left(\tau \delta \right)}$$={\sum }_{\upsilon =0}^{\infty }{\mathcal{B}}_{\upsilon ,q}\left(\tau \right){\displaystyle \frac{{\delta }^{\upsilon }}{{\left[\upsilon \right]}_q!}}$	${\mathcal{A}}_q\left(\delta \right)={\displaystyle \frac{\delta }{e_q\left(\delta \right)-1}}$
II.	The q-EP [3,19]	${\displaystyle \frac{{\left[2\right]}_q}{e_q\left(\delta \right)+1}{e}_q\left(\tau \delta \right)}$$={\sum }_{\upsilon =0}^{\infty }{\mathcal{E}}_{\upsilon ,q}\left(\tau \right){\displaystyle \frac{{\delta }^{\upsilon }}{{\left[\upsilon \right]}_q!}}$	${\mathcal{A}}_q\left(\delta \right)={\displaystyle \frac{{\left[2\right]}_q}{e_q\left(\delta \right)+1}}$
III.	The q-GP [3,19]	${\displaystyle \frac{{\left[2\right]}_q\delta }{e_q\left(\delta \right)+1}{e}_q\left(\tau \delta \right)}$$={\sum }_{\upsilon =0}^{\infty }{\mathcal{G}}_{\upsilon ,q}\left(\tau \right){\displaystyle \frac{{\delta }^{\upsilon }}{{\left[\upsilon \right]}_q!}}$	${\mathcal{A}}_q\left(\delta \right)={\displaystyle \frac{{\left[2\right]}_q\delta }{e_q\left(\delta \right)+1}}$

Appell polynomials of the $r^{th}$ order, which are based on a two-variable q-truncated exponential function and are represented as ${}_{e^{\left(r\right)}}A_{n,q}(\tau ,\varphi )$, are characterized by the generating function provided in [6].

$$A_q\left(t\right){\displaystyle \frac{e_q\left(\tau t\right)}{1-\gamma t^r}}=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}A_{n,q}(\tau ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(28)

The higher-order q-truncated exponential Gould–Hopper polynomials in three variables, symbolized as ${}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$, are described by the following generating function:

$${\displaystyle \frac{e_q\left(\tau t\right)e_q\left(\varphi t^m\right)}{1-\gamma t^r}}=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(29)

In [14], Raza et al. introduced the q-Gould–Hopper polynomials and examined various properties associated with them. Subsequently, in [20], the same authors defined the two-variable q-Gould–Hopper–Appell polynomials by convolving the q-Gould–Hopper polynomials with the q-Appell polynomials, utilizing an extended application of the monomial principle.

The classical Appell polynomial families and their q-analogs have garnered significant scholarly attention due to their applications in combinatorics, quantum theory, and operator analysis. Foundational studies, such as those conducted by Andrews, Askey, and Roy [1], Srivastava and Manocha [4], and Ernst [3], offer a comprehensive framework within which the current generalization can be contextualized.

In this paper, we define three-variable q-truncated exponential Gould–Hopper-based Appell polynomials, employing a generating function approach that incorporates q-calculus functions. The investigation extends to the analysis of these polynomials within the quasi-monomiality framework, facilitating the establishment of crucial operational identities. Subsequently, operational representations are derived, and both q-differential and partial differential equations are formulated for the aforementioned polynomials. To further elucidate their analytical properties, summation formulas are developed. The article concludes with graphical representations that demonstrate the behavior of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials and highlights their potential applications in various fields.

2. q-Appell Convolution

In this section, we introduce three-variable q-truncated exponential Gould–Hopper-based Appell polynomials or, shortly, 3VqTEGHbAPs denoted by ${}_{e^{\left(r\right)}}HA_{n,q}(\tau ,\varphi ,\gamma )$, and we obtain their series definitions, partial derivatives, and explicit and implicit formulas. Therefore, we first define the 3VqTEGHbAPs.

Here, we define the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials in the following form:

$${\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right)=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(30)

In other words, we note that

$$e_q\left(\varphi {\widehat{D}}_{q,\tau }^m\right){\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(31)

We observe that the generating function exhibits absolute convergence for $\left|t\right|<R$, where R is contingent upon the convergence radii of $A_q\left(t\right)$, $e_q\left(\tau t\right)$, and $e_q\left(\varphi t^m\right)$. In this study, we assume $q\in (0,1)$ to ensure the validity of the q-calculus operations and convergence.

Theorem 1. 

For the 3VqTEGHbAP, the following identity holds:

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\sum _{k=0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{A}_{k,q}{}_{e^{\left(r\right)}}H_{n-k,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

(32)

with $A_{k,q}$ given by Equation (27).

Proof. 

By virtue of Equations (29) and (30), we obtain

$$\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}=A_q\left(t\right)\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(33)

Utilizing the expansion (27) of $A_q\left(t\right)$ on the left-hand side of Equation (33), we simplify the expression and equate the coefficients of the corresponding powers of t on both sides of the equation, thereby establishing assertion (32). □

In a similar manner, we can derive the following equivalent forms of the series representations of ${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$.

Theorem 2. 

The three-variable q-truncated exponential Gould–Hopper-based Appell polynomials are characterized by the following series:

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\left[n\right]}_q!\sum _{k,m=0}^{k+sm\le n}{\displaystyle \frac{A_{k,q}{\varphi }^m{E}_{n-k-sm,q}^{\left(r\right)}(\tau ,\gamma )}{{\left[k\right]}_q!{\left[m\right]}_q!{[n-k-sm]}_q!}},$$

(34)

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\left[n\right]}_q!\sum _{k,m=0}^{k+rm\le n}{\displaystyle \frac{A_{k,q}{\gamma }^m{H}_{n-k-rm,q}^{\left(s\right)}(\tau ,\varphi )}{{\left[k\right]}_q!{[n-k-rm]}_q!}},$$

(35)

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\left[n\right]}_q!\sum _{k,m=0}^{rm+sp\le n}{\displaystyle \frac{{\gamma }^m{\varphi }^p{A}_{n-rm-sp,q}^{\left(s\right)}\left(\tau \right)}{{[n-rm-sp]}_q!}}.$$

(36)

Proof. 

In view of Equations (20) and (27)–(29), we can easily obtain (34)–(36). This completes the proof of the theorem. □

The determinant form plays a crucial role in deriving the exploration of q-special polynomials, offering a compact and refined representation that captures their core attributes. This formulation adeptly illustrates essential features, such as orthogonality, recurrence relations, and generating functions, establishing it as a valuable instrument for analysis and manipulation across diverse mathematical settings. Moreover, the determinant form serves as a link to other mathematical frameworks, enabling a deeper understanding of the foundational principles that govern q-special polynomials. Its importance extends beyond mere computational applications, contributing significantly to the theoretical progress in the field of special function theory.

Costabile and Longo [21], expanding on the techniques presented in [22], effectively derived the determinantal form of q-Appell polynomials.

Theorem 3. 

The determinant form of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomial with degree n is

$${}_{e^{\left(r\right)}}H{A}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\displaystyle \frac{1}{{\beta }_{0,q}}},$$

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\displaystyle \frac{{(-1)}^n}{{\left({\beta }_{0,q}\right)}^{n+1}}}$$

$$\times \left|\begin{array}{cccccc}1& {}_{e^{\left(r\right)}}H_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& \dots & {}_{e^{\left(r\right)}}H_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ {\beta }_{0,q}& {\beta }_{1,q}& {\beta }_{2,q}& \dots & {\beta }_{n-1,q}& {\beta }_{n,q}\\ 0& {\beta }_{0,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\beta }_{1,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\beta }_{n-2,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)}_q{\beta }_{n-1,q}\\ 0& 0& {\beta }_{0,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\beta }_{n-3,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)}_q{\beta }_{n-2,q}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & {\beta }_{0,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right)}_q{\beta }_{1,q}\end{array}\right|,$$

(37)

$${\beta }_{n,q}=-{\displaystyle \frac{1}{A_{0,q}}}\left(\sum _{k=1}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{A}_{k,q}{\beta }_{n-k,q}\right),n=0,1,2,\dots ,$$

where ${\beta }_{0,q}\ne 0,$${\beta }_{0,q}={\displaystyle \frac{1}{A_{0,q}}}$ and ${}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),n=0,1,2,\dots ,$ are the three-variable q-truncated exponential Gould–Hopper polynomials defined by Equation (29).

Proof. 

By integrating the series expressions of the newly generalized q-truncated exponential Gould–Hopper polynomials into the generating function of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials, we derive

$$A_q\left(t\right)\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(38)

Multiplying both sides by

$${\displaystyle \frac{1}{A_q\left(t\right)}}=\sum _{\gamma =0}^{\infty }{\beta }_{\gamma ,q}{\displaystyle \frac{t^{\gamma }}{{\left[\gamma \right]}_q!}},$$

(39)

we get

$$\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}=\sum _{\gamma =0}^{\infty }{\beta }_{\gamma ,q}{\displaystyle \frac{{\delta }^{\gamma }}{{\left[\gamma \right]}_q!}}\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(40)

Applying the Cauchy product in (40) gives

$${}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\sum _{\gamma =0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{\gamma }}\right)}_q{\beta }_{\gamma ,q}{}_{e^{\left(r\right)}}H{A}_{n-\gamma ,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ).$$

(41)

So, we obtain the system of equations as follows:

$${}_{e^{\left(r\right)}}H_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\beta }_{0,q}{}_{e^{\left(r\right)}}H{A}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

$${}_{e^{\left(r\right)}}H_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\beta }_{0,q}{}_{e^{\left(r\right)}}H{A}_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+{\beta }_{1,q}{}_{e^{\left(r\right)}}H{A}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

$${}_{e^{\left(r\right)}}H_{2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\beta }_{0,q}{}_{e^{\left(r\right)}}H{A}_{2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+{\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\beta }_{1,q}{}_{e^{\left(r\right)}}H{A}_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+{\beta }_2{}_{e^{\left(r\right)}}H{A}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

                         ⋮

$${}_{e^{\left(r\right)}}H_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\beta }_{0,q}{}_{e^{\left(r\right)}}H{A}_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+{\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\beta }_{1,q}{}_{e^{\left(r\right)}}H{A}_{n-2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+\dots$$

$$+{\beta }_{n-1,q}{}_{e^{\left(r\right)}}H{A}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

$${}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\beta }_{0,q}{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)}_q{\beta }_{1,q}{}_{e^{\left(r\right)}}H{A}_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+\dots$$

$$+{\beta }_{n,q}{}_{e^{\left(r\right)}}H{A}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ).$$

Applying Cramers’ rule, we get

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\displaystyle \frac{\left|\begin{array}{ccccc}{\beta }_{0,q}& 0& \dots & 0& {}_{e^{\left(r\right)}}H_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ {\beta }_{1,q}& {\beta }_{0,q}& \dots & 0& {}_{e^{\left(r\right)}}H_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ {\beta }_{2,q}& {\left(\genfrac{}{}{0pt}{}{2}{1}\right)}_q{\beta }_{1,q}& \dots & 0& {}_{e^{\left(r\right)}}H_{2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ {\beta }_{3,q}& {\left(\genfrac{}{}{0pt}{}{3}{2}\right)}_q{\beta }_{2,q}& \dots & 0& {}_{e^{\left(r\right)}}H_{3,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ ⋮& ⋮& ⋮& ⋮& ⋮& \\ {\beta }_{n-1,q}& {\left(\genfrac{}{}{0pt}{}{n-1}{1}\right)}_q{\beta }_{n-2,q}& \dots & {\beta }_{0,q}& {}_{e^{\left(r\right)}}H_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ {\beta }_{n,q}& {\left(\genfrac{}{}{0pt}{}{n}{1}\right)}_q\beta n-1,q& \dots & {\left(\genfrac{}{}{0pt}{}{n}{n-1}\right)}_q{\beta }_{1,q}& {}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\end{array}\right|}{\left|\begin{array}{ccccc}{\beta }_{0,q}& 0& \dots & 0& 0\\ {\beta }_{1,q}& {\beta }_{0,q}& \dots & 0& 0\\ {\beta }_{2,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\beta }_{1,q}& \dots & 0& 0\\ {\beta }_{3,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2}}\right)}_q{\beta }_{2,q}& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& \\ {\beta }_{n-1,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\beta }_{n-2,q}& \dots & {\beta }_{0,q}& 0\\ {\beta }_{n,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)}_q{\beta }_{n-1,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right)}_q{\beta }_{1,q}& {\beta }_{0,q}\end{array}\right|.}}$$

By taking the transpose in the last equation, we have

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\displaystyle \frac{1}{{\left({\beta }_{0,q}\right)}^{n+1}}}$$

$$\times \left|\begin{array}{ccccc}{\beta }_{0,q}& {\beta }_{1,q}& \dots & {\beta }_{n-1,q}& {\beta }_{n,q}\\ 0& {\beta }_{0,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\beta }_{n-2,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)}_q{\beta }_{n-1,q}\\ 0& 0& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\beta }_{n-3,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)}_q{\beta }_{n-2,q}\\ ⋮& ⋮& ⋮& ⋮& ⋮& \\ 0& 0& \dots & {\beta }_{0,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right)}_q{\beta }_{1,q}\\ {}_{e^{\left(r\right)}}H_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& \dots & {}_{e^{\left(r\right)}}H_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\end{array}\right|.$$

Thus, simple row operations are used to finish the proof. □

Theorem 4. 

The following q-recurrence formula for the 3VqTEGHbAPs holds true:

$${}_{e^{\left(r\right)}}H{A}_{n+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\tau {}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )+{\left[n\right]}_q!\varphi \sum _{m=0}^{n-1}{\displaystyle \frac{{}_{e^{\left(r\right)}}H{A}_{n-m+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )}{{[n-m+1]}_q!}}{T}_{(\varphi ;m)}{T}_{\tau }$$

$$+\sum _{k=0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{}_{e^{\left(r\right)}}H{A}_{n-k,q}^{\left(m\right)}(q\tau ,q\varphi ,\gamma )A_{k+1,q}+{\left[r\right]}_q{\left[n\right]}_q!\sum _{k=0}^{\left[{\displaystyle \frac{n-r+1}{r}}\right]}{\displaystyle \frac{q^{n-kr-r+1}{\gamma }^{k+1}{}_{e^{\left(r\right)}}H{A}_{n-kr-r+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )}{{[n-kr-r+1]}_q!}}.$$

(42)

Proof. 

By differentiating Equation (30) with respect to t using the q-partial derivative and by integrating Equation (15), we derive

$$\begin{array}{c}\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\widehat{D}}_{q,t}{\displaystyle \frac{t^n}{{\left[n\right]}_q!}}={\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{\widehat{D}}_{q,t}\left(e_q\left(\tau t\right)e_q\left(\varphi t^m\right)\right)+{\displaystyle \frac{e_q\left(q\tau t\right)e_q\left(q\varphi t^m\right)}{1-\gamma t^r}}{\widehat{D}}_{q,t}{A}_q\left(t\right)\hfill \\ \hfill +A_q\left(qt\right)e_q\left(q\tau t\right)e_q\left(q\varphi t^m\right){\widehat{D}}_{q,t}{\displaystyle \frac{1}{1-\gamma t^r}}.\end{array}$$

Again, by using Equations (15), (16), (23), and (25) in the above equation, we obtain

$$\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^{n-1}}{{[n-1]}_q!}}=\tau {\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right)+\varphi t^{m-1}{T}_{(\varphi ;m)}{\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(q\tau t\right)e_q\left(\varphi t^m\right)$$

$$+{\displaystyle \frac{e_q\left(q\tau t\right)e_q\left(q\varphi t^m\right)}{1-\gamma t^r}}{\widehat{D}}_{q,t}{A}_q\left(t\right)+\left({\displaystyle \frac{{\left[r\right]}_q\gamma t^{r-1}}{1-\gamma t^r}}\right)\left({\displaystyle \frac{A_q\left(qt\right)}{1-\gamma q^r{t}^r}}{e}_q\left(q\tau t\right)e_q\left(q\varphi t^m\right)\right).$$

(43)

Using the series in the r.h.s of Equation (39), we get

$${\widehat{D}}_{q,t}{A}_q\left(t\right)=\sum _{n=0}^{\infty }{A}_{n+1,q}{\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(44)

Again, using Equations (30) and (44) in Equation (43), we achieve

$$\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}$$

$$=\tau \sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}+{\left[n\right]}_q!\varphi \sum _{n=0}^{\infty }\sum _{m=0}^{n-1}{\displaystyle \frac{{}_{e^{\left(r\right)}}H{A}_{n-m+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )}{{[n-m+1]}_q!}}{T}_{(\varphi ;m)}{T}_{\tau }{\displaystyle \frac{t^n}{{\left[n\right]}_q!}}$$

$$+\sum _{n=0}^{\infty }\sum _{k=0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{}_{e^{\left(r\right)}}H{A}_{n-k,q}^{\left(m\right)}(q\tau ,q\varphi ,\gamma )A_{k+1,q}{\displaystyle \frac{t^n}{{\left[n\right]}_q!}}$$

$$+{\left[r\right]}_q\sum _{n=0}^{\infty }{\left[n\right]}_q!\sum _{k=0}^{\left[{\displaystyle \frac{n-r+1}{r}}\right]}{\displaystyle \frac{q^{n-kr-r+1}{\gamma }^{k+1}{}_{e^{\left(r\right)}}H{A}_{n-kr-r+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )}{{[n-kr-r+1]}_q!}}{\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(45)

By rearranging the series and equating the corresponding powers of t on both sides of the resulting equation, we derive assertion (42). □

Theorem 5. 

The higher derivative formula for the 3VqTEGHbAPs holds true:

$${\widehat{D}}_{q,t}^k{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\displaystyle \frac{{\left[n\right]}_q!}{{[n-k]}_q!}}{}_{e^{\left(r\right)}}H{A}_{n-k,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ).$$

(46)

Proof. 

By taking the $k^{th}$-order q-partial derivative with respect to $\tau$ on both sides of the generating function (30) and subsequently applying Equation (13) to the left-hand side, it can be deduced that

$$\sum _{n=0}^{\infty }{\widehat{D}}_{q,t}^k{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}=t^k{\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right)=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^{n+k}}{{\left[n\right]}_q!}}.$$

(47)

Upon reorganizing the given series and then equating the corresponding powers of t on both sides of the resulting equation, we confirm assertion (30). □

Theorem 6. 

The 3VqTEGHbAPs satisfy the following addition formula:

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau +\delta ,\varphi ,\gamma )=\sum _{k=0}^w{\left({\displaystyle \genfrac{}{}{0pt}{}{w}{k}}\right)}_q{\delta }^k{}_{e^{\left(r\right)}}H{A}_{w-k,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ).$$

(48)

Proof. 

By changing $\tau$ by $\tau +\delta$ in (30) and then using Equation (7), we get

$$A_q\left(t\right){\displaystyle \frac{e_q\left(\tau t\right)e_q\left(\varphi t^m\right)}{1-\gamma t^r}}{e}_q\left(\delta t\right)=\sum _{w=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{w,q}^{\left(m\right)}(\tau +\delta ,\varphi ,\gamma ){\displaystyle \frac{t^w}{{\left[w\right]}_q!}}.$$

(49)

Using Equations (7) and (30) in l.h.s. of the above equation, we have

$$\sum _{w=0}^{\infty }\sum _{k=0}^{\infty }{\delta }^k{}_{e^{\left(r\right)}}H{A}_{w,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^{w+k}}{{\left[w\right]}_q!{\left[k\right]}_q!}}=\sum _{w=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{w,q}^{\left(m\right)})(\tau +\delta ,\varphi ,\gamma ){\displaystyle \frac{t^w}{{\left[w\right]}_q!}}.$$

Through the process of rearranging the series and equating the corresponding powers of t on each side, assertion (48) is substantiated. □

Theorem 7. 

The summation formula for the ${}_{e^{\left(r\right)}}H{A}_{w,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ is explicitly defined as the product of $H_{w,q}^{\left(m\right)}(\tau ,\varphi )$ and $e_{w,q}^{\left(r\right)}(\tau ,\varphi )$, as demonstrated below:

$${}_{e^{\left(r\right)}}H{A}_{w,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\sum _{s=0}^w\sum _{k=0}^{w-s}{\left({\displaystyle \genfrac{}{}{0pt}{}{w}{s}}\right)}_q{\left({\displaystyle \genfrac{}{}{0pt}{}{w-s}{k}}\right)}_q{(-w)}^s{H}_{w-k-s}^{\left(m\right)}(\tau ,\gamma )e_{k,q}^{\left(r\right)}(w,\varphi ).$$

(50)

Proof. 

We examine the product formed by the generating function (20) of $H_{w,q}^{\left(m\right)}(\tau ,\varphi )$ and generating function (28) of ${}_{e^{\left(r\right)}}A_{w,q}(\tau ,\varphi )$ in the following form:

$$A_q\left(t\right){\displaystyle \frac{e_q\left(wt\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^r\right)=\sum _{w=0}^{\infty }\sum _{k=0}^{\infty }{H}_{w,q}^{\left(m\right)}(\tau ,\gamma ){}_{e^{\left(r\right)}}A_{k,q}(\tau ,\varphi ){\displaystyle \frac{t^{w+k}}{{\left[w\right]}_q!{\left[k\right]}_q!}}.$$

(51)

In the r.h.s. of Equation (51), using the identity

$$\sum _{w=0}^{\infty }\sum _{k=0}^{\infty }A(k,w)=\sum _{w=0}^{\infty }\sum _{k=0}^{w}A(k,w-k),$$

(52)

we get

$$A_q\left(t\right){\displaystyle \frac{e_q\left(wt\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^r\right)=\sum _{w=0}^{\infty }\sum _{k=0}^w{H}_{w-k,q}^{\left(m\right)}(\tau ,\gamma ){}_{e^{\left(r\right)}}A_{k,q}(\tau ,\varphi ){\displaystyle \frac{t^{w+k}}{{[w-k]}_q!{\left[k\right]}_q!}},$$

(53)

which, on shifting the $e_q\left(wt\right)$ to the l.h.s. and using the series definition of the exponential, becomes

$${\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^r\right)=\sum _{w=0}^{\infty }\sum _{s=0}^{\infty }\sum _{k=0}^w{\left({\displaystyle \genfrac{}{}{0pt}{}{w}{k}}\right)}_q{(-w)}^s{H}_{w-k,q}^{\left(m\right)}(\tau ,\gamma )e_{k,q}^{\left(r\right)}(w,\varphi ){\displaystyle \frac{t^{w+s}}{{\left[w\right]}_q!{\left[s\right]}_q!}}.$$

(54)

Again, using Equation (28) in the r.h.s of Equation (54), we get

$${\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^r\right)=\sum _{w=0}^{\infty }\sum _{s=0}^w\sum _{k=0}^{w-s}{\left({\displaystyle \genfrac{}{}{0pt}{}{w-s}{k}}\right)}_q{(-w)}^s{H}_{w-k-s,q}^{\left(m\right)}(\tau ,\gamma ){}_{e^{\left(r\right)}}A_{k,q}(\tau ,\varphi ){\displaystyle \frac{t^w}{{[w-s]}_q!{\left[s\right]}_q!}}.$$

(55)

Utilizing the equation generating function (30) on the left-hand side of Equation (55) and matching the coefficients of identical powers of t in the resulting equation, we establish assertion (50). □

3. Monomiality Characteristic and Operational Identities

This section presents the q-quasi-monomiality characteristic, operational identities, and q-differential equations pertinent to the 3VqTEGHbAPs, denoted as ${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$.

The notion of monomiality is a crucial tool for analyzing exceptional polynomials and their attributes. Initially introduced by J.F. Steffensen [23], this concept was later expanded into the realm of quasi-monomiality by Dattoli and his colleagues [24,25,26]. In the specific context of q-polynomials, Raza et al. [15] further extended the monomiality principle. This extension offers a robust methodology for examining the quasi-monomiality of certain q-special polynomials. Researchers have extensively utilized monomiality frameworks to construct and evaluate hybrid families of special polynomials [27]. The two q-operators, denoted as ${\widehat{M}}_q$ and ${\widehat{P}}_q$, are known as the q-multiplicative and q-derivative operators, respectively, for a q-polynomial set $p_{n,q}\left(\tau \right)(n\in \mathbb{N},\tau \in \mathbb{C})$, as demonstrated by [2]

$$\widehat{M_q}\left\{p_{n,q}\left(\tau \right)\right\}=p_{n+1,q}\left(\tau \right),$$

(56)

and

$$\widehat{P_q}\left\{p_{n,q}\left(\tau \right)\right\}={\left[n\right]}_q{p}_{n-1,q}\left(\tau \right).$$

(57)

The operators $\widehat{M_q}$ and $\widehat{P_q}$ adhere to the following commutation relation:

$$[\widehat{M_q},\widehat{P_q}]=\widehat{P_q}\widehat{M_q}-\widehat{M_q}\widehat{P_q}.$$

(58)

The characteristics of the polynomials $p_{n,q}\left(\tau \right)$ can be inferred from the properties of the operators $\widehat{M_q}$ and $\widehat{P_q}$. If these operators, $\widehat{M_q}$ and $\widehat{P_q}$, have a differential realization, then the polynomials $p_{n,q}\left(\tau \right)$ are governed by the differential equation

$$\widehat{M_q}\widehat{P_q}\left\{p_{n,q}\left(\tau \right)\right\}={\left[n\right]}_q{p}_{n,q}\left(\tau \right),$$

(59)

and

$$\widehat{P_q}\widehat{M_q}\left\{p_{n,q}\left(\tau \right)\right\}={[n+1]}_q{p}_{n,q}\left(\tau \right).$$

(60)

In view of (56) and (57), we have

$$[\widehat{M_q},\widehat{P_q}]={[n+1]}_q-{\left[n\right]}_q.$$

(61)

From (56), we have

$${\widehat{M_q}}^r\left\{p_{n,q}\right\}=p_{n+r,q}\left(\tau \right).$$

(62)

In particular, we have

$$p_{n,q}\left(\tau \right)={\widehat{M_q}}^n\left\{p_{0,q}\right\}={\widehat{M_q}}^n\left\{1\right\},$$

(63)

where $p_{0,q}\left(\tau \right)=1$ is the q-sequel of polynomial $p_{n,q}\left(\tau \right)$. Also, the generating function of $p_{n,q}\left(\tau \right)$ can be obtained as

$$e_q\left(\widehat{M_q}t\right)\left\{1\right\}=\sum _{n=0}^{\infty }{p}_{n,q}\left(\tau \right){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(64)

Now, we establish the q-monomial characteristic of the three-variable q-truncated Gould–Hopper–Appell polynomials in the form of the following theorem.

Theorem 8. 

The three-variable q-truncated exponential Gould–Hopper-based Appell polynomials exhibit quasi-monomial properties when subjected to the following q-multiplicative and q-derivative operators:

$${\widehat{M}}_{3VqTGAP}=\tau +\varphi {\widehat{D}}_{q,\tau }^{m-1}{T}_{(\varphi ;m)}{T}_{\tau }+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^{r-1}}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi },$$

(65)

or, alternatively,

$${\widehat{M}}_{3VqTGAP}=\tau T_{\varphi }+\varphi {\widehat{D}}_{q,\tau }^{m-1}{T}_{(\varphi ;m)}+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^{r-1}}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi },$$

(66)

and

$${\widehat{P}}_{3VqTGAP}={\widehat{D}}_{q,\tau },$$

(67)

respectively, where $T_{\tau }$ and $T_{\varphi }$ are the q-dilatation operators, as specified in Equation (23).

Proof. 

By taking the partial derivative of Equation (30) with respect to t and applying Equation (15), we derive

$$\sum _{n=1}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\widehat{D}}_{q,t}{\displaystyle \frac{t^n}{{\left[n\right]}_q!}}$$

$$={\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{\widehat{D}}_{q,t}\left(e_q\left(\tau t\right)e_q\left(\varphi t^m\right)\right)+{\displaystyle \frac{e_q\left(q\tau t\right)e_q\left(q\varphi t^m\right)}{1-\gamma t^r}}{\widehat{D}}_{q,t}{A}_q\left(t\right)+A_q\left(qt\right)e_q\left(q\tau t\right)e_q\left(q\varphi t^m\right){\widehat{D}}_{q,t}{\displaystyle \frac{1}{1-\gamma t^r}}.$$

(68)

By employing Equations (15) and (16) and setting $f_q\left(t\right)=e_q\left(\varphi t\right)$ and $g_q\left(t\right)=e_q\left(\varphi t^m\right)$, followed by simplifying the resulting equation using Equations (23) and (25) on the left-hand side, we obtain

$$\sum _{n=1}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^{n-1}}{{[n-1]}_q!}}$$

$$=\left(\tau +\varphi t^{m-1}{T}_{(\varphi ;m)}{T}_{\tau }+{\displaystyle \frac{A_q^{\prime }\left(t\right)}{A_q\left(t\right)}}{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma t^{r-1}}{1-q^r\gamma t^r}}\right)\left({\displaystyle \frac{A_q\left(qt\right)}{A_q\left(t\right)}}\right)T_{\tau }{T}_{\varphi }\right){\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right).$$

(69)

Since

$${\widehat{D}}_{q,\tau }{\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right)=t{\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right).$$

(70)

$A_q\left(t\right)$ is an invertible series of t, and ${\displaystyle \frac{A_q^{\prime }\left(t\right)}{A_q\left(t\right)}}$ has q-power series expansion in t.

Therefore, using Equations (69) and (70), we get

$$\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}$$

$$=\left(\tau +\varphi {\widehat{D}}_{q,\tau }^{m-1}{T}_{(\varphi ;m)}{T}_{\tau }+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^{r-1}}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi }\right)$$

$$\times {\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right),$$

(71)

which, on using (30), gives

$$\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{A}_{n+1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}$$

$$=\sum _{n=0}^{\infty }\left(\tau +\varphi {\widehat{D}}_{q,\tau }^{m-1}{T}_{(\varphi ;m)}{T}_{\tau }+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^{r-1}}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi }\right)$$

$$\times {}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(72)

By equating the coefficients of t on both sides of Equation (72) and considering Equation (56), the resulting equation substantiates assertion (65).

By utilizing Equation (68) and defining $f_q\left(t\right)=e_q\left(\varphi t^m\right)$ alongside $g_q\left(t\right)=e_q\left(\varphi t\right)$ and by replicating the procedural steps detailed in the proof of Equation (65), we confirm assertion (66).

In view of (57), we note that Equation (46) (for $k=1$) proves assertion (67). □

Theorem 9. 

The following q-differential equations for ${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ hold true:

$$\left(\tau {\widehat{D}}_{q,\tau }+\varphi {\widehat{D}}_{q,\tau }^m{T}_{(\varphi ;m)}{T}_{\tau }+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{\widehat{D}}_{q,\tau }{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^r}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi }-{\left[n\right]}_q\right)$$

$$\times {}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0,$$

(73)

and

$$\left(\tau {\widehat{D}}_{q,\tau }{T}_{\varphi }+\varphi {\widehat{D}}_{q,\tau }^m{T}_{(\varphi ;m)}+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{\widehat{D}}_{q,\tau }{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^r}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi }-{\left[n\right]}_q\right)$$

$$\times {}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0.$$

(74)

Proof. 

Using (65), (66), and (67) in (59), we get

$$\left(\tau {\widehat{D}}_{q,\tau }+\varphi {\widehat{D}}_{q,\tau }^m{T}_{(\varphi ;m)}{T}_{\tau }+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{\widehat{D}}_{q,\tau }{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^r}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi }\right)$$

$$\times {}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\left[n\right]}_q{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

(75)

and

$$\left(\tau {\widehat{D}}_{q,\tau }{T}_{\varphi }+\varphi {\widehat{D}}_{q,\tau }^m{T}_{(\varphi ;m)}+{\displaystyle \frac{A_q^{\prime }\left({\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}{\widehat{D}}_{q,\tau }{T}_{\tau }{T}_{\varphi }+\left({\displaystyle \frac{{\left[r\right]}_q\gamma {\widehat{D}}_{q,\tau }^r}{1-q^r\gamma {\widehat{D}}_{q,\tau }^r}}\right)\left({\displaystyle \frac{A_q\left(q{\widehat{D}}_{q,\tau }\right)}{A_q\left({\widehat{D}}_{q,\tau }\right)}}\right)T_{\tau }{T}_{\varphi }\right)$$

$$\times {}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={\left[n\right]}_q{}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

(76)

Therefore, upon simplification, we get the assertions (73) and (74). □

The application of the monomiality principle in this context inherently results in the formulation of canonical q-differential operators, for which the polynomials serve as eigenfunctions. This structural characteristic implies that each polynomial family under consideration can be systematically linked to a q-difference equation, thereby reinforcing its analytical and spectral importance. We present the following theorem regarding the operational identities associated with the three-variable q-truncated Gould–Hopper polynomials.

Theorem 10. 

The 3VqTEGHbAPs satisfy the following respective operational identities:

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=e_q\left(\varphi {\widehat{D}}_{q,\tau }^m\right)\left\{{}_{e^{\left(r\right)}}A_{n,q}(\tau ,\gamma )\right\},$$

(77)

or, equivalently,

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi +l,\gamma )=e_q\left(l{\widehat{D}}_{q,\tau }^m\right)\left\{{}_{e^{\left(r\right)}}A_{n,q}(\tau ,\gamma )\right\},$$

(78)

and

$$E_q\left(-\varphi {\widehat{D}}_{q,\tau }^m\right){}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={}_{e^{\left(r\right)}}A_{n,q}(\tau ,\gamma ),$$

(79)

where ${\widehat{D}}_{q,\tau }^m$ is the $m^{th}$ q-derivative operator.

Proof. 

In view of Equation (7), we have

$${\widehat{D}}_{q,\tau }^m{\tau }^w={\displaystyle \frac{{\left[w\right]}_q!}{{[w-mr]}_q!}}{\tau }^{w-mr}.$$

(80)

Utilizing the preceding equation of Formula (30), we acquire

$${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\varphi {\widehat{D}}_{q,\tau }^m\right)}^k}{{\left[k\right]}_q!}}{}_{e^{\left(r\right)}}A_{n,q}(\tau ,\gamma ).$$

(81)

Utilizing expression (5) on the r.h.s. of the preceding equation, we arrive at statement (77). Again, by using a similar method to (77) and (5), we get assertion (78). By operating $E_q\left(-\varphi {\widehat{D}}_{q,\tau }^m\right)$ on both sides of Equation (77) and using Equation (10), we obtain (79). □

4. Reductions to Classical Families

This section presents specific cases of the polynomial previously discussed in Section 2, represented by Equation (30). Taking $A_q\left(t\right)={\displaystyle \frac{t}{e_q\left(t\right)-1}}$ in (30), ${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ reduces to the q-truncated exponential Gould–Hopper-based Bernoulli polynomials (q-TEGHbBPs); ${}_{e^{\left(r\right)}}H{B}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ are defined by the following generating function:

$${\displaystyle \frac{t}{e_q\left(t\right)-1}}{\displaystyle \frac{e_q\left(\tau t\right)e_q\left(\varphi t^m\right)}{1-\gamma t^r}}=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{B}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(82)

In view of Equations (22) and (82), we have

$${}_{e^{\left(r\right)}}H{B}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=e_q\left(\varphi {\widehat{D}}_{q,\tau }^m\right)\left\{{}_{e^{\left(r\right)}}B_{n,q}(\tau ,\gamma )\right\}.$$

(83)

Utilizing Equation (29), we proceed to expand the left-hand side of Equation (82), and we have

$${}_{e^{\left(r\right)}}H{B}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\sum _{k=0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{B}_{k,q}{}_{e^{\left(r\right)}}H_{n-k,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ).$$

(84)

Considering Equation (82), the following determinant form for ${}_{e^{\left(r\right)}}H{B}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ is derived:

Theorem 11. 

The determinant representation of q-truncated Gould–Hopper–Bernoulli polynomials of degree n is

$${}_{e^{\left(r\right)}}H{B}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=1,$$

$${}_{e^{\left(r\right)}}H{B}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={(-1)}^n$$

$$\times \left|\begin{array}{cccccc}1& {}_{e^{\left(r\right)}}H_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& \dots & {}_{e^{\left(r\right)}}H_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ 1& {\displaystyle \frac{1}{{\left[2\right]}_q}}& {\displaystyle \frac{1}{{\left[3\right]}_q}}& \dots & {\displaystyle \frac{1}{{\left[n\right]}_q}}& {\displaystyle \frac{1}{{[n+1]}_q}}\\ 0& 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\displaystyle \frac{1}{{\left[2\right]}_q}}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\displaystyle \frac{1}{{[n-1]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)}_q{\displaystyle \frac{1}{{\left[n\right]}_q}}\\ 0& 0& 1& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\displaystyle \frac{1}{{[n-2]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)}_q{\displaystyle \frac{1}{{[n-1]}_q}}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right)}_q{\displaystyle \frac{1}{{\left[2\right]}_q}}\end{array}\right|,$$

(85)

where ${}_{e^{\left(r\right)}}H{B}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ ($n=1,2,\dots$) and ${}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),n=0,1,2,\dots ,$ are the q-truncated Gould–Hopper polynomials defined by Equation (29).

Furthermore, by taking $A_q\left(t\right)={\displaystyle \frac{2}{e_q\left(t\right)+1}}$ in (30), ${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ reduces to the q-truncated exponential Gould–Hopper-based Euler polynomials (q-TEGHbEPs); ${}_{e^{\left(r\right)}}H{E}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ are defined by the following generating function:

$${\displaystyle \frac{2}{e_q\left(t\right)+1}}{\displaystyle \frac{e_q\left(\tau t\right)e_q\left(\varphi t^m\right)}{1-\gamma t^r}}=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{E}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(86)

By virtue of (22) and (60), we have

$${}_{e^{\left(r\right)}}H{E}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=e_q\left(\varphi {\widehat{D}}_{q,\tau }^m\right)\left\{{}_{e^{\left(r\right)}}E_{n,q}(\tau ,\gamma )\right\}.$$

(87)

Expanding the left-hand side of Equation (86) by using Equation (29), we have

$${}_{e^{\left(r\right)}}H{E}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\sum _{k=0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{E}_{k,q}{}_{e^{\left(r\right)}}H_{n-k,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

(88)

The following determinant form for ${}_{e^{\left(r\right)}}H{E}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ is obtained.

Theorem 12. 

The determinant formulation of the q-truncated Gould–Hopper–Euler polynomials for a polynomial of degree n is

$${}_{e^{\left(r\right)}}H{E}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=1,$$

$${}_{e^{\left(r\right)}}H{E}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={(-1)}^n$$

$$\times \left|\begin{array}{cccccc}1& {}_{e^{\left(r\right)}}H_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& \dots & {}_{e^{\left(r\right)}}H_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ 1& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& \dots & {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ 0& 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\displaystyle \frac{1}{2}}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\displaystyle \frac{1}{2}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)}_q{\displaystyle \frac{1}{2}}\\ 0& 0& 1& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\displaystyle \frac{1}{2}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)}_q{\displaystyle \frac{1}{2}}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right)}_q{\displaystyle \frac{1}{2}}\end{array}\right|,$$

(89)

where ${}_{e^{\left(r\right)}}H{E}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ ($n=1,2,\dots$) and ${}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),n=0,1,2,\dots ,$ are the q-truncated Gould–Hopper polynomials defined by Equation (29).

By taking $A_q\left(t\right)={\displaystyle \frac{2t}{e_q\left(t\right)+1}}$ in (30), ${}_{e^{\left(r\right)}}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ is reduced to the q-truncated exponential Gould–Hopper-based Genocchi polynomials (q-TEGHbGPs); ${}_{e^{\left(r\right)}}H{G}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ are defined by the following generating function:

$${\displaystyle \frac{2t}{e_q\left(t\right)+1}}{\displaystyle \frac{e_q\left(\tau t\right)e_q\left(\varphi t^m\right)}{1-\gamma t^r}}=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{G}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{{\left[n\right]}_q!}}.$$

(90)

In view of Equations (22) and (64), we have

$${}_{e^{\left(r\right)}}H{G}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=e_q\left(\varphi {\widehat{D}}_{q,\tau }^m\right)\left\{{}_{e^{\left(r\right)}}G_{n,q}(\tau ,\gamma )\right\}.$$

(91)

Expanding the left-hand side of Equation (65) by using Equation (29), we have

$${}_{e^{\left(r\right)}}H{G}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=\sum _{k=0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{G}_{k,q}{}_{e^{\left(r\right)}}H_{n-k,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),$$

(92)

Theorem 13. 

The determinant representation of the q-truncated exponential Gould–Hopper-based Genocchi polynomials (q-TEGHbGPs) ${}_{e^{\left(r\right)}}H{G}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ of degree n is

$${}_{e^{\left(r\right)}}H{G}_{0,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=1,$$

$${}_{e^{\left(r\right)}}H{G}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )={(-1)}^n$$

$$\times \left|\begin{array}{cccccc}1& {}_{e^{\left(r\right)}}H_{1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{2,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& \dots & {}_{e^{\left(r\right)}}H_{n-1,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )& {}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )\\ 1& {\displaystyle \frac{1}{2{\left[2\right]}_q}}& {\displaystyle \frac{1}{2{\left[3\right]}_q}}& \dots & {\displaystyle \frac{1}{2{\left[n\right]}_q}}& {\displaystyle \frac{1}{2{[n+1]}_q}}\\ 0& 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\displaystyle \frac{1}{2{\left[2\right]}_q}}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\displaystyle \frac{1}{2{[n-1]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right)}_q{\displaystyle \frac{1}{2{\left[n\right]}_q}}\\ 0& 0& 1& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right)}_q{\displaystyle \frac{1}{2{[n-2]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)}_q{\displaystyle \frac{1}{2{[n-1]}_q}}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right)}_q{\displaystyle \frac{1}{2{\left[2\right]}_q}}\end{array}\right|,$$

(93)

where ${}_{e^{\left(r\right)}}H{G}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$ ($n=1,2,\dots$) and ${}_{e^{\left(r\right)}}H_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma ),n=0,1,2,\dots ,$ are the q-truncated exponential Gould–Hopper polynomials defined by Equation (29).

Remark 1. 

To further enhance clarity and confirm consistency with classical results, it is instructive to consider the limiting case as $q\to 1$. In this limit, the q-truncated exponential Gould–Hopper-based Appell polynomials reduce to their classical counterparts, including the Bernoulli, Euler, and Genocchi polynomials.

For instance, taking the generating function of the q-truncated exponential Gould–Hopper-based Bernoulli polynomials with

$$A_q\left(t\right)={\displaystyle \frac{t}{e_q\left(t\right)-1}},$$

and considering the classical limit $q\to 1$, we obtain

$$\underset{q\to 1}{lim}\left({\displaystyle \frac{t}{e_q\left(t\right)-1}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right){\displaystyle \frac{1}{1-\gamma t^r}}\right)={\displaystyle \frac{t}{e^t-1}}{e}^{\tau t}{e}^{\varphi t^m}{\displaystyle \frac{1}{1-\gamma t^r}}.$$

This expression corresponds to the generating function of the classical truncated exponential Gould–Hopper-based Bernoulli polynomials:

$${\displaystyle \frac{t}{e^t-1}}{e}^{\tau t}{e}^{\varphi t^m}{\displaystyle \frac{1}{1-\gamma t^r}}=\sum _{n=0}^{\infty }{}_{e^{\left(r\right)}}H{B}_n^{\left(m\right)}(\tau ,\varphi ,\gamma ){\displaystyle \frac{t^n}{n!}}.$$

Similar derivations can be performed for the Euler and Genocchi polynomials.

Remark 2 

(On Time-Reversal Symmetry). A critical yet frequently neglected aspect in the analysis of special polynomials is their behavior under the inversion of the dependent variable, specifically the transformation $t\mapsto -t$. This consideration of symmetry is of considerable importance in physical and engineering contexts, where it often corresponds to time reversibility or irreversibility in dynamical systems. For example, even functions $p\left(t\right)=p(-t)$ exhibit time-symmetric behavior, whereas odd or asymmetric functions indicate directional or irreversible processes.

In the case of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials $e^{\left(r\right)}H{A}_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$, a study of the generating function

$${\displaystyle \frac{A_q\left(t\right)}{1-\gamma t^r}}{e}_q\left(\tau t\right)e_q\left(\varphi t^m\right)$$

under the substitution $t\mapsto -t$ could yield insights into the parity structure of the polynomials and their coefficients. Such an investigation could further distinguish between even, odd, or asymmetrically structured polynomial families, as well as elucidate deeper algebraic or physical symmetries embedded in the q-calculus framework.

Remark 3 

(On Canonical Differential Equations). The introduction of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials in this study, along with the formulation of canonical q-differential equations, is both feasible and significant. By utilizing the generating function structure in conjunction with the monomiality principle and operational identities, one can systematically derive q-difference or q-differential equations for which these polynomials act as eigenfunctions. These equations not only elucidate the analytical structure of the polynomials but also facilitate applications in mathematical physics and integrable systems.

5. Distribution of Zeros and Graphical Representation

In this section, we intend to present the graphical representations and zeros of the q-truncated exponential Gould–Hopper-based Bernoulli polynomials ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )$, as delineated in Section 3. For $m=3$, a few of ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )$ are as follows:

$$\begin{array}{ll}{}_{e^{\left(4\right)}}HB_{0,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& 1,\\ {}_{e^{\left(4\right)}}HB_{1,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& \tau -{\displaystyle \frac{1}{{\left[2\right]}_q!}},\\ {}_{e^{\left(4\right)}}HB_{2,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -\tau +{\tau }^2+{\displaystyle \frac{1}{{\left[2\right]}_q!}}-{\displaystyle \frac{{\left[2\right]}_q!}{{\left[3\right]}_q!}},\\ {}_{e^{\left(4\right)}}HB_{3,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -\tau +{\tau }^3+{\displaystyle \frac{2}{{\left[2\right]}_q!}}+\varphi {\left[3\right]}_q!-{\displaystyle \frac{{\left[3\right]}_q!}{{\left[2\right]}_q{!}^3}}+{\displaystyle \frac{\tau {\left[3\right]}_q!}{{\left[2\right]}_q^2}}-{\displaystyle \frac{{\tau }^2{\left[3\right]}_q!}{{\left[2\right]}_q{!}^2}}-{\displaystyle \frac{{\left[3\right]}_q!}{{\left[4\right]}_q!}},\\ {}_{e^{\left(4\right)}}HB_{4,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -\tau +{\tau }^4+{\displaystyle \frac{2}{{\left[2\right]}_q!}}+\gamma {\left[4\right]}_q!+\tau \varphi {\left[4\right]}_q!+{\displaystyle \frac{{\left[4\right]}_q!}{{\left[2\right]}_q{!}^4}}-{\displaystyle \frac{\tau {\left[4\right]}_q!}{{\left[2\right]}_q{!}^3}}\\ & +{\displaystyle \frac{{\tau }^2{\left[4\right]}_q!}{{\left[2\right]}_q{!}^3}}-{\displaystyle \frac{\varphi {\left[4\right]}_q!}{{\left[2\right]}_q!}}+{\displaystyle \frac{{\left[4\right]}_q!}{{\left[3\right]}_q{!}^2}}-{\displaystyle \frac{3{\left[4\right]}_q!}{{\left[2\right]}_q{!}^2{\left[3\right]}_q!}}+{\displaystyle \frac{2\tau {\left[4\right]}_q!}{{\left[2\right]}_q!{\left[3\right]}_q!}}\\ & -{\displaystyle \frac{{\tau }^2{\left[4\right]}_q!}{{\left[2\right]}_q!{\left[3\right]}_q!}}-{\displaystyle \frac{{\tau }^3{\left[4\right]}_q!}{{\left[2\right]}_q!{\left[3\right]}_q!}}-{\displaystyle \frac{{\left[4\right]}_q!}{{\left[5\right]}_q!}},\end{array}$$

$$\begin{array}{ll}{}_{e^{\left(4\right)}}HB_{5,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -\tau +{\tau }^5+{\displaystyle \frac{2}{{\left[2\right]}_q!}}+\gamma \tau {\left[5\right]}_q!-{\displaystyle \frac{{\left[5\right]}_q!}{{\left[2\right]}_q{!}^5}}+{\displaystyle \frac{\tau {\left[5\right]}_q!}{{\left[2\right]}_q^4}}-{\displaystyle \frac{{\tau }^2{\left[5\right]}_q!}{{\left[2\right]}_q{!}^4}}\\ & +{\displaystyle \frac{\varphi {\left[5\right]}_q!}{{\left[2\right]}_q{!}^2}}-{\displaystyle \frac{\gamma {\left[5\right]}_q!}{{\left[2\right]}_q!}}-{\displaystyle \frac{\tau \varphi {\left[5\right]}_q!}{{\left[2\right]}_q!}}+{\displaystyle \frac{{\tau }^2\varphi {\left[5\right]}_q!}{{\left[2\right]}_q!}}+{\displaystyle \frac{\tau {\left[5\right]}_q!}{{\left[3\right]}_q{!}^2}}-{\displaystyle \frac{{\tau }^3{\left[5\right]}_q!}{{\left[3\right]}_q^2}}\\ & -{\displaystyle \frac{3{\left[5\right]}_q!}{{\left[2\right]}_q!{\left[3\right]}_q{!}^2}}-{\displaystyle \frac{\varphi {\left[5\right]}_q!}{{\left[3\right]}_q!}}+{\displaystyle \frac{4{\left[5\right]}_q!}{{\left[2\right]}_q{!}^3{\left[3\right]}_q!}}-{\displaystyle \frac{3\tau {\left[5\right]}_q!}{{\left[2\right]}_q{!}^2{\left[3\right]}_q!}}\\ & +{\displaystyle \frac{2{\tau }^2{\left[5\right]}_q!}{{\left[2\right]}_q{!}^2{\left[3\right]}_q!}}+{\displaystyle \frac{{\tau }^3{\left[5\right]}_q!}{{\left[2\right]}_q{!}^2{\left[3\right]}_q!}}-{\displaystyle \frac{3{\left[5\right]}_q!}{{\left[2\right]}_q{!}^2{\left[4\right]}_q!}}+{\displaystyle \frac{2\tau {\left[5\right]}_q!}{{\left[2\right]}_q!{\left[4\right]}_q!}}\\ & -{\displaystyle \frac{{\tau }^2{\left[5\right]}_q!}{{\left[2\right]}_q!{\left[4\right]}_q!}}-{\displaystyle \frac{{\tau }^4{\left[5\right]}_q!}{{\left[2\right]}_q!{\left[4\right]}_q!}}+{\displaystyle \frac{2{\left[5\right]}_q!}{{\left[3\right]}_q!{\left[4\right]}_q!}}-{\displaystyle \frac{{\left[5\right]}_q!}{{\left[6\right]}_q!}}.\end{array}$$

We conduct an investigation into the zeros of the q-truncated exponential Gould–Hopper-based Bernoulli polynomials, denoted as ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$, utilizing computational methods. Specifically, we plot the zeros of the q-truncated Hopper–Bernoulli polynomials ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ for $n=30$ (see Figure 1).

Figure 1. Zeros of ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

In Figure 1 (top left), the parameters are set as follows: $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{3}{10}}$. In Figure 1 (top right), the parameters are $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{5}{10}}$. In Figure 1 (bottom left), the parameters are $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{7}{10}}$. In Figure 1 (bottom right), the parameters are $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{9}{10}}$.

Stacks of zeros of the q-truncated Hopper–Bernoulli polynomials ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ for $1\le n\le 30$, forming a 3D structure, are presented in Figure 2.

Figure 2. Zeros of ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

In Figure 2 (top left), the parameters are set as follows: $m=3,r=4,\varphi =2,\gamma =7,$ and $q={\displaystyle \frac{3}{10}}$. In Figure 2 (top right), the parameters are $m=3,r=4,\varphi =2,\gamma =7,$ and $q={\displaystyle \frac{5}{10}}$. In Figure 2 (bottom left), the parameters are $m=3,r=4,\varphi =2,\gamma =7,$ and $q={\displaystyle \frac{7}{10}}$. In Figure 2 (bottom right), the parameters are $m=3,r=4,\varphi =2,\gamma =7,$ and $q={\displaystyle \frac{9}{10}}$.

Plots of real zeros of q-truncated exponential Gould–Hopper-based Bernoulli polynomials ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ for $1\le n\le 30$ are presented in Figure 3.

Figure 3. Real zeros of ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

In Figure 3 (top left), the parameters are set as follows: $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{3}{10}}$. In Figure 3 (top right), the parameters are $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{5}{10}}$. In Figure 3 (bottom left), the parameters are $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{7}{10}}$. In Figure 3 (bottom right), the parameters are $m=3$, $r=4$, $\varphi =2$, $\gamma =7$, and $q={\displaystyle \frac{9}{10}}$.

Subsequently, we computed an approximate solution that satisfies the q-truncated exponential Gould–Hopper-based Bernoulli polynomials ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ for the parameters $m=3,r=4,\varphi =2,\gamma =7,$ and $q={\displaystyle \frac{7}{10}}$. The results are presented in Table 2.

Table 2. Approximate solutions of ${}_{e^{\left(r\right)}}HB_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

Degree n	$\mathit{\tau }$
1	0.58824
2	0.15593, 0.84407
3	−1.5676, 1.4279 − 1.6511i, 1.4279 + 1.6511i
4	−1.6771 − 1.3048i, −1.6771 + 1.3048i, 2.4221 − 2.5091i, 2.4221 + 2.5091i
5	−2.3820 − 1.9547i, −2.3820 + 1.9547i, 0.60620, 2.8945 − 3.1030i, 2.8945 + 3.1030i
6	−2.8833 − 2.3341i, −2.8833 + 2.3341i, 0.54216 − 0.65641i, 0.54216 + 0.65641i, 3.2062 − 3.4847i, 3.2062 + 3.4847i
7	−3.1692 − 2.6645i, −3.1692 + 2.6645i, −1.4251, 1.3897 − 1.7140i, 1.3897 + 1.7140i, 3.3918 − 3.6791i, 3.3918 + 3.6791i
8	−3.3242 − 2.7625i, −3.3242 + 2.7625i, −1.6097 − 1.6085i, −1.6097 + 1.6085i, 2.5029 − 2.5017i, 2.5029 + 2.5017i, 3.3549 − 3.7042i, 3.3549 + 3.7042i
9	−3.4373 − 2.5476i, −3.4373 + 2.5476i, −2.2913 − 2.6109i, −2.2913 + 2.6109i, 0.56410, 2.8826 − 3.8504i, 2.8826 + 3.8504i, 3.5048 − 2.9482i, 3.5048 + 2.9482i
10	−3.8254 − 2.3390i, −3.8254 + 2.3390i, −2.4606 − 3.2866i, −2.4606 + 3.2866i, 0.50785 − 0.65764i, 0.50785 + 0.65764i, 2.7588 − 4.2451i, 2.7588 + 4.2451i, 3.9721 − 2.9318i, 3.9721 + 2.9318i

For $m=3$, a few of ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )$ are as follows:

$$\begin{array}{ll}{}_{e^{\left(4\right)}}HE_{0,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& 1,\\ {}_{e^{\left(4\right)}}HE_{1,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -{\displaystyle \frac{1}{2}}+\tau ,\\ {}_{e^{\left(4\right)}}HE_{2,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -{\displaystyle \frac{1}{2}}+{\tau }^2+{\displaystyle \frac{1}{4}}{\left[2\right]}_q!-{\displaystyle \frac{1}{2}}\tau {\left[2\right]}_q!,\\ {}_{e^{\left(4\right)}}HE_{3,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -{\displaystyle \frac{1}{2}}+{\tau }^3-{\displaystyle \frac{1}{8}}{\left[3\right]}_q!+{\displaystyle \frac{1}{4}}\tau {\left[3\right]}_q!+\varphi {\left[3\right]}_q!+{\displaystyle \frac{{\left[3\right]}_q!}{2{\left[2\right]}_q!}}-{\displaystyle \frac{\tau {\left[3\right]}_q!}{2{\left[2\right]}_q!}}-{\displaystyle \frac{{\tau }^2{\left[3\right]}_q!}{2{\left[2\right]}_q!}},\\ {}_{e^{\left(4\right)}}HE_{4,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -{\displaystyle \frac{1}{2}}+{\tau }^4+{\displaystyle \frac{1}{16}}{\left[4\right]}_q!+\gamma {\left[4\right]}_q!-{\displaystyle \frac{1}{8}}\tau {\left[4\right]}_q!\\ & -{\displaystyle \frac{1}{2}}\varphi {\left[4\right]}_q!+\tau \varphi {\left[4\right]}_q!+{\displaystyle \frac{{\left[4\right]}_q!}{4{\left[2\right]}_q{!}^2}}-{\displaystyle \frac{{\tau }^2{\left[4\right]}_q!}{2{\left[2\right]}_q{!}^2}}\\ & -{\displaystyle \frac{3{\left[4\right]}_q!}{8{\left[2\right]}_q!}}+{\displaystyle \frac{\tau {\left[4\right]}_q!}{2{\left[2\right]}_q!}}+{\displaystyle \frac{{\tau }^2{\left[4\right]}_q!}{4{\left[2\right]}_q!}}+{\displaystyle \frac{{\left[4\right]}_q!}{2{\left[3\right]}_q!}}-{\displaystyle \frac{\tau {\left[4\right]}_q!}{2{\left[3\right]}_q!}}-{\displaystyle \frac{{\tau }^3{\left[4\right]}_q!}{2{\left[3\right]}_q!}},\\ {}_{e^{\left(4\right)}}HE_{5,q}^{\left(3\right)}(\tau ,\varphi ,\gamma )=& -{\displaystyle \frac{1}{2}}+{\tau }^5-{\displaystyle \frac{1}{32}}{\left[5\right]}_q!-{\displaystyle \frac{1}{2}}\gamma {\left[5\right]}_q!+{\displaystyle \frac{1}{16}}\tau {\left[5\right]}_q!+\gamma \tau {\left[5\right]}_q!\\ & +{\displaystyle \frac{1}{4}}\varphi {\left[5\right]}_q!-{\displaystyle \frac{1}{2}}\tau \varphi {\left[5\right]}_q!-{\displaystyle \frac{3{\left[5\right]}_q!}{8{\left[2\right]}_q^2}}+{\displaystyle \frac{\tau {\left[5\right]}_q!}{4{\left[2\right]}_q^2}}+{\displaystyle \frac{{\tau }^2\left[5\right]}{2{\left[2\right]}_q{!}^2}}+{\displaystyle \frac{{\left[5\right]}_q!}{4{\left[2\right]}_q!}}\\ & -{\displaystyle \frac{3\tau {\left[5\right]}_q!}{8{\left[2\right]}_q!}}-{\displaystyle \frac{{\tau }^2{\left[5\right]}_q!}{8{\left[2\right]}_q!}}-{\displaystyle \frac{\varphi {\left[5\right]}_q!}{2{\left[2\right]}_q!}}+{\displaystyle \frac{{\tau }^2\varphi {\left[5\right]}_q!}{{\left[2\right]}_q!}}-{\displaystyle \frac{3{\left[5\right]}_q!}{8{\left[3\right]}_q!}}+{\displaystyle \frac{\tau {\left[5\right]}_q!}{2{\left[3\right]}_q!}}\\ & +{\displaystyle \frac{{\tau }^3{\left[5\right]}_q!}{4{\left[3\right]}_q!}}+{\displaystyle \frac{{\left[5\right]}_q!}{2{\left[2\right]}_q!{\left[3\right]}_q!}}-{\displaystyle \frac{{\tau }^2{\left[5\right]}_q!}{2{\left[2\right]}_q!{\left[3\right]}_q!}}-{\displaystyle \frac{{\tau }^3{\left[5\right]}_q!}{2{\left[2\right]}_q!{\left[3\right]}_q!}}+{\displaystyle \frac{{\left[5\right]}_q!}{2{\left[4\right]}_q!}}\\ & -{\displaystyle \frac{\tau {\left[5\right]}_q!}{2{\left[4\right]}_q!}}-{\displaystyle \frac{{\tau }^4{\left[5\right]}_q!}{2{\left[4\right]}_q!}}.\end{array}$$

We investigate the beautiful zeros of the q-truncated exponential Gould–Hopper-based Euler polynomials ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ by using a computer. We plot the zeros of q-truncated Hopper–Euler polynomials ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ for $n=30$ (Figure 4).

Figure 4. Zeros of ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

In Figure 4 (top left), the parameters are set as follows: $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{3}{10}}$. In Figure 4 (top right), the parameters are $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{5}{10}}$. In Figure 4 (bottom left), the parameters are $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{7}{10}}$. In Figure 4 (bottom right), the parameters are $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{9}{10}}$.

Figure 5 illustrates the three-dimensional structure formed by the zeros of the q-truncated Hopper–Euler polynomials ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ for $1\le n\le 30$.

Figure 5. Zeros of ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

In Figure 5 (top left), the selected parameters are $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{3}{10}}$. In Figure 5 (top right), the parameters remain as $m=3$, $r=4$, $\varphi =5$, and $\gamma =10$, with q adjusted to ${\displaystyle \frac{5}{10}}$. In Figure 5 (bottom left), the parameters are consistent with $m=3$, $r=4$, $\varphi =5$, and $\gamma =10$, and q is set to ${\displaystyle \frac{7}{10}}$. Finally, in Figure 5 (bottom right), the parameters are $m=3$, $r=4$, $\varphi =5$, and $\gamma =10$, and q is increased to ${\displaystyle \frac{9}{10}}$.

Plots of real zeros of q-truncated Hopper–Euler polynomials ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$ for $1\le n\le 30$ are presented in Figure 6.

Figure 6. Real zeros of ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

In Figure 6 (top left), the parameters are set as follows: $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{3}{10}}$. In Figure 6 (top right), the parameters are $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{5}{10}}$. In Figure 6 (bottom left), the parameters are $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{7}{10}}$. In Figure 6 (bottom right), the parameters are $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{9}{10}}$.

Subsequently, we computed an approximate solution that satisfies the q-truncated exponential Gould–Hopper-based Euler polynomials ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$, with the parameters set as $m=3$, $r=4$, $\varphi =5$, $\gamma =10$, and $q={\displaystyle \frac{7}{10}}$. The results are presented in Table 3.

Table 3. Approximate solutions of ${}_{e^{\left(r\right)}}HE_{n,q}^{\left(m\right)}(\tau ,\varphi ,\gamma )=0$.

Degree n	$\mathit{\tau }$
1	0.50000
2	−0.080594, 0.93059
3	−2.3549, 1.7249 − 2.2326i, 1.7249 + 2.2326i
4	−2.0245 − 0.1768i, −2.0245 + 0.1768i, 2.6578 − 3.1756i, 2.6578 + 3.1756i
5	−2.7749 − 1.3907i, −2.7749 + 1.3907i, 0.56863, 3.1838 − 3.8462i, 3.1838 + 3.8462i
6	−3.4673 − 1.9184i, −3.4673 + 1.9184i, 0.6617 − 1.3704i, 0.6617 + 1.3704i, 3.5409 − 4.2425i, 3.5409 + 4.2425i
7	−3.7243 − 2.3058i, −3.7243 + 2.3058i, −1.8144, 1.6543 − 2.4440i, 1.6543 + 2.4440i, 3.7418 − 4.4043i, 3.7418 + 4.4043i
8	−3.9250 − 2.4629i, −3.9250 + 2.4629i, −1.6504 − 1.3176i, −1.6504 + 1.3176i, 2.6170 − 3.2519i, 2.6170 + 3.2519i, 3.7436 − 4.3541i, 3.7436 + 4.3541i
9	−4.1243 − 2.4363i, −4.1243 + 2.4363i, −2.1590 − 2.2507i, −2.1590 + 2.2507i, 0.22580, 3.0404 − 4.4562i, 3.0404 + 4.4562i, 3.9297 − 3.7151i, 3.9297 + 3.7151i
10	−4.4008 − 2.2964i, −4.4008 + 2.2964i, −2.6341 − 2.9269i, −2.6341 + 2.9269i, 0.4638 − 1.4799i, 0.4638 + 1.4799i, 2.9738 − 4.9105i, 2.9738 + 4.9105i, 4.4070 − 3.5880i, 4.4070 + 3.5880i

6. Conclusions

The concept of q-monomiality plays a pivotal role in the analysis of special polynomials, providing a framework for understanding their properties and relationships. This paper delves into a comprehensive examination of three-variable q-truncated exponential Gould–Hopper-based Appell polynomials within the context of q-monomiality. By systematically studying these polynomials, the authors uncover and elucidate various properties, contributing to a deeper understanding of their mathematical structure and behavior. The research extends beyond theoretical exploration, offering practical applications and insights. The authors present special cases of the newly established polynomials, accompanied by their determinantal forms. This approach not only demonstrates the versatility of the polynomials but also provides concrete examples for further study. Additionally, this paper includes an analysis of zeros and graphical representations for these special cases, offering visual and numerical insights into the polynomials’ behavior. These findings serve as a valuable resource for researchers, enabling them to apply the q-monomiality principle to investigate the properties of various polynomials and potentially uncover new mathematical relationships.