A New Generalization of q-Laguerre-Based Appell Polynomials and Quasi-Monomiality

Abstract

In this paper, we define a new generalization of three-variable q-Laguerre polynomials and derive some properties. By using these polynomials, we introduce a new generalization of three-variable q-Laguerre-based Appell polynomials (3VqLbAP) through a generating function approach involving zeroth-order q-Bessel–Tricomi functions. These polynomials are studied by means of generating function, series expansion, and determinant representation. Also, these polynomials are further examined within the framework of q-quasi-monomiality, leading to the establishment of essential operational identities. We then derive operational representations, as well as q-differential equations for the three-variable q-Laguerre-based Appell polynomials. Some examples are constructed in terms of q-Laguerre–Hermite-based Bernoulli, Euler, and Genocchi polynomials in order to illustrate the main results.

1. Introduction

The Laguerre polynomials constitute an important class of orthogonal polynomials with diverse applications across various fields, including quantum group theory, harmonic oscillator theory, and coding theory [1,2,3]. One notable application of these polynomials is their role in expressing covariant oscillator algebra, where they provide a powerful mathematical framework. Dattoli and Torre [4] extensively studied the theory of two-variable Laguerre polynomials, demonstrating that classical Laguerre polynomials can be formulated within the framework of quasi-monomials. The significance of two-variable Laguerre polynomials stems from their profound mathematical properties, as they naturally emerge as solutions to certain partial differential equations, including the heat diffusion equation. Furthermore, they play a crucial role in radiation physics, particularly in problems involving electromagnetic wave propagation and quantum beam lifetime in storage rings [5]. Their broad applicability underscores their importance in both theoretical and applied mathematics.

In recent years, the study of q-calculus and fractional q-differential equations has gained significant momentum among mathematicians and engineers due to its profound applications in various scientific fields. Originally introduced by Jackson’s in 1908, q-calculus, often referred to as quantum calculus, provides a framework for defining and analyzing q-differential equations [6,7]. These equations play a crucial role in modeling discrete physical processes, including those found in quantum mechanics, dynamical systems, and stochastic analysis. Typically, q-differential equations are formulated on a discrete time scale $T_q$, where q represents the scaling parameter [8,9,10,11]. As the theory of q-calculus has evolved, several fundamental extensions have emerged, such as the q-Laplace transform, q-Gamma and q-Beta functions, q-Mittag–Leffler functions, and q-integral transforms. Although q-calculus has seen remarkable theoretical advancements, the study of fractional q-calculus remains relatively underdeveloped compared to its classical counterpart, highlighting the need for further research and exploration [12,13,14].

In this work, we assume that $0<q<1$, and we adhere to the terminolgy and notions given in [15,16]. The q-shift factorial ${(\alpha ;q)}_{\theta }$ is defined by

$${(\alpha ;q)}_{\theta }=\prod _{s=1}^{\theta -1}(1-q^s\alpha ),\theta \in \mathbb{N},\alpha \in \mathbb{C};0<q<1.$$

(1)

The q-numbers and q-factorial are described as follows:

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

(2)

and

$${\left[\theta \right]}_q!=\left\{\begin{array}{ccc}{\prod }_{\nu =1}^{\theta }{\left[\nu \right]}_q,& 0<q<1,\theta \ne 0\in \mathbb{C}& \\ & & \\ 1,& \theta =0.& \end{array}\right..$$

(3)

The ${(\mu \pm a)}_q^{\theta }$ (or Gauss’s q-binomial) formula can be shown by the following [15,16]:

$${(\mu \pm a)}_q^{\theta }=\sum _{\nu =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\nu }}\right)}_q{q}^{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -s}{2}}\right)}{\mu }^{\nu }{(\pm a)}^{\theta -\nu }.$$

(4)

The two type q-exponential functions are described by the following [17,18,19]:

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

(5)

and

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

(6)

The product of both q-exponential functions are given by the following:

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

(7)

Consequently,

$$e_q(\gamma )E_q(-\gamma )=e_q(-\gamma )E_q(\gamma )=1,\mid \gamma \mid <{\displaystyle \frac{1}{1-q}}.$$

(8)

The q-derivative operator ${\widehat{D}}_{q,\mu }$ operating on the function $f(\mu )$ is defined as follows [15,16,20]:

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

(9)

The q-derivative of the q-exponential functions are provided by the following:

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

(10)

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

(11)

and

$${\widehat{D}}_{q,\mu }^k{e}_q(\alpha \mu )={\alpha }^k{e}_q(\alpha \mu ),k\in \mathbb{N},\alpha \in \mathbb{C}.$$

(12)

The product of two functions $f(\mu )$ and $g(\mu )$ considered by the following [21,22]:

$${\widehat{D}}_{q,\mu }(f(\mu )g(\mu ))=f(\mu ){\widehat{D}}_{q,\mu }g(\mu )+g(q\mu ){\widehat{D}}_{q,\mu }f(\mu ).$$

(13)

Recently, Cao et al. [21] introduced the 2-variable q-Laguerre polynomials $L_{n,q}(\mu ,\eta )$ described by

$$C_{0,q}(\mu \delta )e_q(\eta \delta )=\sum _{\theta =0}^{\infty }{L}_{\theta ,q}(\mu ,\eta ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(14)

where $C_{0,q}(\mu )$ is the (0^th order q-Bessel–Tricomi) provided by

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

(15)

The operational identity of q-Laguerre polynomials $L_{\theta ,q}(\mu ,\eta )$ is given by the following (see [21]):

$$e_q(-{\widehat{D}}_{q,\mu }^{-1}t)e_q(\eta \delta )\left\{1\right\}=\sum _{\theta =0}^{\infty }{L}_{\theta ,q}(\mu ,\eta ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(16)

Individually, for $m\in \mathbb{N}\cup \left\{1\right\}$ and supposing ${\widehat{D}}_{q,\mu }^{-1}\left\{1\right\}=\mu$, it is noted that

$${\left({\widehat{D}}_{q,\mu }^{-1}\right)}^m\left\{1\right\}={\displaystyle \frac{{\mu }^m}{{\left[m\right]}_q!}}.$$

(17)

The q-derivative of the 0^th order Tricomi functions $C_{0,q}(\mu \delta )$ are given as follows:

$${\widehat{D}}_{q,\mu }\mu {\widehat{D}}_{q,\mu }{C}_{0,q}(\mu \delta )={\displaystyle \frac{{\partial }_q}{{\partial }_q{\widehat{D}}_{q,\mu }^{-1}}}{C}_{0,q}(\mu \delta )=-\delta C_{0,q}(\mu \delta ),$$

(18)

and

$${\widehat{D}}_{q,\mu }\mu {\widehat{D}}_{q,\mu }{C}_{0,q}(\alpha \mu \delta )={\displaystyle \frac{{\partial }_q}{{\partial }_q{\widehat{D}}_{q,\mu }^{-1}}}{C}_{0,q}(\alpha \mu \delta ).$$

(19)

The q-Hermite polynomials $H_{\theta ,q}(\mu ,\eta )$ are defined by means of the subsequent generating function [23],

$$e_q(\mu \delta )e_q(\eta {\delta }^2)=\sum _{\theta =0}^{\infty }{H}_{\theta ,q}(\mu ,\eta ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(20)

and series definition,

$$H_{\theta ,q}(\mu ,\eta )={\left[\theta \right]}_q!\sum _{k=0}^{\left[{\displaystyle \frac{\theta }{2}}\right]}{\displaystyle \frac{y^k{\mu }^{\theta -2k}}{{\left[k\right]}_q!{[\theta -2k]}_q!}}.$$

(21)

The q-dilatation operator $T_{\mu }$, operating on any complex variable function $f(\mu )$ in the following manner, is [21]

$$T_{\mu }^{k}f(\mu )=f(q^k\mu ),k\in \mathbb{R},0<q<1,$$

(22)

where

$$T_{\mu }^{-1}{T}_{\mu }^{1}f(\mu )=f(\mu ).$$

(23)

The q-derivative operator ${\widehat{D}}_{q,\delta }$ operating on $e_q(\eta {\delta }^m)$ provided by the following [21]:

$${\widehat{D}}_{q,\delta }{e}_q(\eta {\delta }^m)=\eta {\delta }^{m-1}{T}_{(\eta ;m)}{e}_q(\eta {\delta }^m),$$

(24)

In the previous equation, the following condition exists:

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

Al Salam introduced and defined q-Appell polynomials ${\mathcal{A}}_{n,q}(x)$ as follows [24,25]:

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

(25)

where

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

(26)

The monomiality concept is a valuable technique for understanding certain special polynomials and functions in conjunction with relations and properties. This concept has the potential to generate novel sets of the family of q-special polynomials and show the quasi-monomial nature of some previously established q-extension of special polynomials. For further information, one may look at the works of [22,26,27,28]. Implementing the monomiality principle to quantum calculus establishes a basis for comprehending q-special polynomials as specific solutions to expanded versions of q-integro differential equations and q-partial differential equations. Cao and colleagues [21] have recently expanded the idea of the monomiality principle to the field of quantum calculus.

Let $s_{\theta ,q}(\mu )$ be a q-polynomial set for $(\theta \in \mathbb{N}$ and $\mu \in \mathbb{C})$. The q-multiplicative operator ${\widehat{M}}_q$ and q-derivative operator ${\widehat{P}}_q$ are provided as follows [21]:

$$\widehat{M_q}\left\{s_{\theta ,q}(\mu )\right\}=s_{\theta +1,q}(\mu ),$$

(27)

and

$$\widehat{P_q}\left\{s_{\theta ,q}(\mu )\right\}={\left[\theta \right]}_q{s}_{\theta -1,q}(\mu ),$$

(28)

which fulfill the following relation:

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

(29)

The characteristic of the polynomials $s_{\theta ,q}(\mu )$ can be deduced from the features of the $\widehat{M_q}$ and $\widehat{P_q}$, which have a q-differential realization; then, the polynomials $s_{\theta ,q}(\mu )$ must satisfy the q-differential equation

$$\widehat{M_q}\widehat{P_q}\left\{s_{\theta ,q}(x)\right\}={\left[\theta \right]}_q{s}_{\theta ,q}(\mu ),$$

(30)

and

$$\widehat{P_q}\widehat{M_q}\left\{s_{\theta ,q}(\mu )\right\}={[\theta +1]}_q{s}_{\theta ,q}(\mu ).$$

(31)

In view of (27) and (28), we have

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

(32)

From (27), we have

$${\widehat{M_q}}^r\left\{s_{\theta ,q}\right\}=s_{\theta +r,q}(\mu ).$$

(33)

In particular, we have

$$s_{\theta ,q}(\mu )={\widehat{M_q}}^{\theta }\left\{s_{0,q}\right\}={\widehat{M_q}}^{\theta }\left\{1\right\},$$

(34)

where $s_{0,q}(\theta )=1$. The sequel polynomials $s_{\theta ,q}(\mu )$ can be presented as

$$e_q(\widehat{M_q}\delta )\left\{1\right\}=\sum _{\theta =0}^{\infty }{s}_{\theta ,q}(\mu ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(35)

Recently, Alam and colleague [29] introduced and defined the two-variable q-general polynomials (2VqgP) $p_{\theta ,q}(\zeta ,\eta )$ provided by the following:

$$e_q(\gamma \delta ){\varphi }_q(\nu ,\delta )=\sum _{\theta =0}^{\infty }{p}_{\theta ,q}(\gamma ,\nu ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},(p_{0,q}(\gamma ,\nu )=1),$$

(36)

where ${\varphi }_q(\nu ,\delta )$ has (at least the formal) series expansion as follows:

$${\varphi }_q(\nu ,\delta )=\sum _{\theta =0}^{\infty }{\varphi }_{\theta ,q}(\nu ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},({\varphi }_{0,q}(\nu )\ne 0).$$

(37)

Note that when $q\to 1^{-}$, the Equation (36) reduces to the two variable general polynomials $p_{\theta }(\gamma ,\nu )$ (see [30]).

The q-Laguerre-based Appell polynomials exhibit intriguing connections with symmetry identities, particularly in the realm of combinatorial mathematics and special functions. These polynomials often satisfy recurrence relations and q-difference equations that reveal symmetrical structures when parameters are interchanged. Additionally, their generating functions can display inherent symmetries, making them useful in deriving new transformation formulas. Orthogonality properties further reinforce symmetry by linking polynomial families through integral representations. Such identities play a crucial role in applications involving quantum calculus, q-series, and statistical mechanics, where symmetric structures naturally emerge in physical and mathematical models.

In this paper, we provide and examine the unique features of q-Laguerre-based Appell polynomials with three variables by employing the q-operators. Moreover, we provide applications of these newly discovered q-Laguerre-based Appell polynomials. Overall, our findings suggest that the new generalization of three-variable q-Laguerre-based Appell polynomials have promising applications in various fields.

2. New Generalization of q-Laguerre-Based Appell Polynomials

This part defines the new generalization of three-variable q-Laguerre-based Appell polynomials ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ utilizing the function $C_{0,q}(\mu \delta )$ in (15) and derives explicit formulas, operational identities, q-quasimonomiality characteristic and q-differential equations for these polynomials. We perform to define three-variable q-Laguerre polynomials, referred to as ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ as follows:

Definition 1. 

We define the new generalization of three-variable q-Laguerre polynomials ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ as follows:

$$e_q(\mu \delta ){\varphi }_q(\eta ,\delta )C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},(p_{0,q}(\mu ,\eta )=1),$$

(38)

where $C_{0,q}(\omega \delta )$ and ${\varphi }_q(\eta ,\delta )$ show the summation series in Equations (15) and (37).

Theorem 1. 

The new generalization of three-variable q-Laguerre polynomials ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ are defined by the following series:

$${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{\varphi }_{\gamma ,q}(\eta )L_{\theta -\gamma ,q}(\mu ,\omega ).$$

(39)

Proof. 

Through (14) and (37) in l.h.s of (38), we obtain the assertion of (39). □

Here, we deduce the respective operational identities for the new generalization of three-variable q-Laguerre polynomials ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$.

Theorem 2. 

The new generalization of three-variable q-Laguerre polynomials ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ satisfy the following respective operational identities:

$${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=e_q\left(-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }\right)\left\{p_{\theta ,q}(\mu ,\eta )\right\},$$

(40)

or, equivalently,

$${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=C_{0,q}\left(\omega {\widehat{D}}_{q,\mu }\right)\left\{p_{\theta ,q}(\mu ,\eta )\right\},$$

(41)

and

$$E_q\left({\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\zeta }\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=p_{\theta ,q}(\mu ,\eta ).$$

(42)

Proof. 

In view of Equation (10), we have

$${\widehat{D}}_{q,\mu }^r{\mu }^{\theta }={\displaystyle \frac{{\left[\theta \right]}_q!}{{[\theta -r]}_q!}}{\mu }^{\theta -r}.$$

(43)

Using Equation (38), we obtain

$${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r{\omega }^r{\widehat{D}}_{q,\mu }^r}{{({\left[r\right]}_q!)}^2}}{p}_{\theta ,q}(\mu ,\eta ).$$

(44)

Consider (44) and using (5) and (17), we acquire the assertion (40). Therefore, using (16) and (40), we obtain (41). On multiplying $E_q\left({\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }\right)$ of (40) and then using Equation (8), we acquire the result (42). The proof of Theorem 2 is complete. □

Now, we establish the following theorem, including q-multiplicative and q-derivative operators of ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$.

Theorem 3. 

The new generalization of three-variable q-Laguerre polynomials ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ demonstrate quasimonomial characteristics when subject to the q-multiplicative and q-derivative operators:

$${\widehat{M}}_{3VqLP}=\mu T_{\omega }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1},$$

(45)

or, alternatively,

$${\widehat{M}}_{3VqLP}=\mu +{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu },$$

(46)

and

$${\widehat{P}}_{3VqLP}={\widehat{D}}_{q,\mu },$$

(47)

where $T_{\mu }$ and $T_{\omega }$ denote the q-dilatation operators (see (22)).

Proof. 

By using (15) and applying q-derivative operator of (38) partially with respect to $\delta$, we obtain

$$\sum _{\theta =1}^{\infty }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\widehat{D}}_{q,\delta }{\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}=e_q(q\mu \delta )C_{0,q}(q\omega \delta ){\widehat{D}}_{q,\delta }{\varphi }_q(\eta ,\delta )+{\varphi }_q(\eta ,\delta ){\widehat{D}}_{q,\delta }{e}_q(\mu \delta )C_{0,q}(\omega \delta ).$$

(48)

Using (11), (17), and (18) in l.h.s. of (48), we have

$$\sum _{\theta =1}^{\infty }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta -1}}{{[\theta -1]}_q!}}=e_q(q\mu \delta )C_{0,q}(q\omega \delta ){\varphi }_q^{\prime }(\eta ,\delta )+\mu {\varphi }_q(\eta ,\delta )C_{0,q}(q\omega \delta )e_q(\mu \delta )-{\widehat{D}}_{q,\omega }^{-1}{\varphi }_q(\eta ,\delta )C_{0,q}(\omega \delta )e_q(\mu \delta ).$$

(49)

Again, using (38) and (22) in (49), we acquire

$$\sum _{\theta =0}^{\infty }{}_{p}L_{\theta +1,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}=\sum _{\theta =0}^{\infty }\left(\mu T_{\omega }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,\delta )}{{\varphi }_q(\eta ,\delta )}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(50)

which yields the following assertion:

$${}_{p}L_{\theta +1,q}(\mu ,\eta ,\omega )=\left(\mu T_{\omega }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,\delta )}{{\varphi }_q(\eta ,\delta )}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ).$$

(51)

Thus, with (27), the above equation yields the assertion (45) of Theorem 3.

Again, using Equation (48), by taking $f_q(t)=C_{0,q}(zt)$ and $g_q(\delta )=e_q(\mu \delta )$ and following the same lines of proof of assertion (45), we obtain

$${}_{p}L_{\theta +1,q}(\mu ,\eta ,\omega )=\left(\mu +{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ),$$

(52)

which yields the second assertion result (46).

If we apply the operator ${\widehat{D}}_{q,\mu }$ to the both sides of (38) and employ (11), we have

$$\delta e_q(\mu \delta ){\varphi }_q(\eta ,\delta )C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{\widehat{D}}_{q,\mu }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(53)

By employing (38), we acquire

$${\widehat{D}}_{q,x}\left\{{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )\right\}={\left[\theta \right]}_q{}_{p}L_{\theta -1,q}(\mu ,\eta ,\omega ),(\theta ⩾1),$$

(54)

which, in view of (28), yields the assertion (47) of Theorem 3. □

Theorem 4. 

The following q-differential equations for new generalization of three-variable q-Laguerre polynomials ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ hold true:

$$\left(\mu T_{\omega }{\widehat{D}}_{q,\mu }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }-{\left[\theta \right]}_q\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=0,$$

(55)

and

$$\left(\mu {\widehat{D}}_{q,\mu }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }{\widehat{D}}_{q,\mu }-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }{T}_{\mu }-{\left[\theta \right]}_q\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=0.$$

(56)

Proof. 

Inserting Equations (45) and (46) in (30), it follows that

$$\left(\mu T_{\omega }{\widehat{D}}_{q,\mu }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )={\left[\theta \right]}_q{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ),$$

(57)

and

$$\left(\mu {\widehat{D}}_{q,\mu }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }{\widehat{D}}_{q,\mu }-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }{T}_{\mu }\right){}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )={\left[\theta \right]}_q{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ).$$

(58)

Therefore, after simplification, we obtain the assertions (55) and (56) of Theorem 4. □

Remark 1. 

Since $p_{0,q}(\mu ,\eta )=1$, by utilizing (34), we have

$${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )={\left(\mu T_{\omega }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}\right)}^n\left\{1\right\}(p_{0,q}(\mu ,\eta )=1).$$

Also, in view of Equations (35), (38) and (45), we have

$$e_q\left({\widehat{M}}_{3VqLP}\right)\left\{1\right\}=e_q(\mu \delta ){\varphi }_q(\eta ,\delta )C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(59)

Now, we proceed to introduce the new generalization of three-variable q-Laguerre-based Appell polynomials ($3VqLAP$) in order to derive the generating functions for the new generalization of three-variable q-Laguerre-based Appell polynomials by means of exponential generating function of q-Appell polynomials (25). Thus, replacing $\mu$ on the left hand side of (45) by the q-multiplicative operator of ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ given by (38) denoting the new generalization of three-variable q-Laguerre-based Appell polynomials ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$, we obtain

$$A_q(t)e_q\left({\widehat{M}}_{3VqLP}\right)\left\{1\right\}=\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(60)

which on using Equation (45), we obtain

$$A_q(t)e_q\left(\mu T_{\omega }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}\right)\left\{1\right\}=\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(61)

Using the relation (59) in (60), the polynomials ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ in the following form:

$$A_q(\delta )e_q(\mu \delta ){\varphi }_q(\eta ,\delta )C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(62)

Theorem 5. 

The new generalization of three-variable q-Laguerre-based Appell polynomials ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ are defined by the following series:

$${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{A}_{\gamma ,q}{}_{p}L_{\theta -\gamma ,q}(\mu ,\eta ,\gamma ).$$

(63)

Proof. 

Utilizing (38) and (62), we can write

$$\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}=A_q(\delta )\sum _{\theta =0}^{\infty }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(64)

Substituting the expansion (26) of $A_q(\delta )$ into the l.h.s. of (64), and simplifying, we acquire (63). □

The determinant form is fundamental in the study of q-special polynomials, offering a concise and elegant representation that encapsulates their essential properties. This formulation effectively expresses key characteristics such as orthogonality, recurrence relations, and generating functions, making it a powerful tool for analysis and manipulation in various mathematical contexts. Additionally, the determinant form serves as a bridge to other mathematical structures, facilitating deeper insights into the fundamental principles governing q-special polynomials. Its significance extends beyond practical computations, contributing to the broader theoretical advancements in special function theory.

Keleshteri and Mahmudov [31], building upon the methods introduced in [32,33], successfully derived the determinantal form of q-Appell polynomials. Acknowledging the crucial role of determinant representations in both computational and applied mathematics, they extended their approach to obtain the determinant representation of the three-variable Laguerre-based Appell polynomials, denoted as ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$. This was achieved through the proof of a key result, which establishes a structured and elegant formulation of these polynomials, further enhancing their utility in mathematical analysis and applications.

Theorem 6. 

The determinant representation for the new generalization of three-variable q-Laguerre-based Appell polynomials ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ of degree n is

$${}_{p}L{\mathbb{A}}_{0,q}(\mu ,\eta ,\omega )={\displaystyle \frac{1}{{\beta }_{0,q}}},$$

$${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )={\displaystyle \frac{{(-1)}^{\theta }}{{({\beta }_{0,q})}^{\theta +1}}}\left|\begin{array}{cccccc}1& {}_{p}L_{1,q}(\mu ,\eta ,\omega )& {}_{p}L_{2,q}(\mu ,\eta ,\omega )& \dots & {}_{p}L_{\theta -1,q}(\mu ,\eta ,\omega )& {}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )\\ {\beta }_{0,q}& {\beta }_{1,q}& {\beta }_{2,q}& \dots & {\beta }_{\theta -1,q}& {\beta }_{n,q}\\ 0& {\beta }_{0,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\beta }_{1,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{1}}\right)}_q{\beta }_{\theta -2,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{1}}\right)}_q{\beta }_{\theta -1,q}\\ 0& 0& {\beta }_{0,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{1}}\right)}_q{\beta }_{\theta -3,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{2}}\right)}_q{\beta }_{\theta -2,q}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & {\beta }_{0,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\theta -1}}\right)}_q{\beta }_{1,q}\end{array}\right|,$$

(65)

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

where ${\beta }_{0,q}\ne 0,$ ${\beta }_{0,q}={\displaystyle \frac{1}{A_{0,q}}}$ and ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ),\theta =0,1,2,\dots ,$ are the new generalization of three-variable q-Laguerre polynomials defined by Equation (38).

Proof. 

By substituting the series representations of the newly generalized three-variable q-Laguerre polynomials into the generating function of the three-variable q-Laguerre-based Appell polynomials, we derive

$$A_q(\delta )\sum _{\theta =0}^{\infty }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}=\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(66)

Multiplying both sides by

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

(67)

we obtain

$$\sum _{\theta =0}^{\infty }{}_{p}L_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}=\sum _{\gamma =0}^{\infty }{\beta }_{\gamma ,q}{\displaystyle \frac{{\delta }^{\gamma }}{{\left[\gamma \right]}_q!}}\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(68)

Applying Cauchy product in (68) gives

$${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{\beta }_{\gamma ,q}{}_{p}L{\mathbb{A}}_{\theta -\gamma ,q}(\mu ,\eta ,\omega ).$$

(69)

This equality leads to a system of $\theta$-equations with unknown ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ),\theta =0,1,2,\cdots$.

To solve this system, we employ Cramer’s rule, leveraging the fact that the denominator corresponds to the determinant of a lower triangular matrix, which simplifies to ${\left({\beta }_{0,q}\right)}^{\theta +1}$. By transposing the numerator and systematically shifting the $i^{th}$ row by $(i+1)$-th position, for $i=1,2,\cdots ,\theta -1$, we obtain the desired result in a structured and computationally efficient manner. □

Theorem 7. 

The new generalization of three-variable generalized q-Laguerre-based Appell polynomials ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ possess quasi-monomial properties under the action of the following q-multiplicative and q-derivative operators:

$${\widehat{M}}_{3VqLAP}=\left(\mu +{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}},$$

(70)

or, alternatively

$${\widehat{M}}_{3VqLAP}=\left(\mu T_{\omega }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}},$$

(71)

and

$${\widehat{P}}_{3VqAP}={\widehat{D}}_{q,\mu },$$

(72)

respectively.

Proof. 

Applying the partial q-derivative to the (62) with respect to $\delta$ and employing (15), we acquire

$$\sum _{\theta =1}^{\infty }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\widehat{D}}_{q,\delta }{\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}$$

$$=A_q(q\theta )e_q(q\mu \delta ){\widehat{D}}_{q,\delta }\left(C_{0,q}(\omega \delta ){\varphi }_q(\eta ,\delta )\right)+C_{0,q}(\omega \delta ){\varphi }_q(\eta ,\delta ){\widehat{D}}_{q,\delta }\left(e_q(\mu \delta )A_q(\delta )\right).$$

(73)

By (15), (17) and (22) in the r.h.s. of (73), we have

$$\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta +1,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

$$=\left(\left(\mu +{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,\delta )}{{\varphi }_q(\eta ,\delta )}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q\delta )}{A_q(\delta )}}+{\displaystyle \frac{A_q^{\prime }(\delta )}{A_q(\delta )}}\right)A_q(t)e_q(\mu \delta ){\varphi }_q(\eta ,\delta )C_{0,q}(\omega \delta ),$$

(74)

which, in view of Equation (62), becomes

$$\sum _{\theta =0}^{\infty }{}_{p}L{\mathbb{A}}_{\theta +1,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

$$=\sum _{\theta =0}^{\infty }\left(\left(\mu +{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}\right){}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(75)

which yields from (75), we obtain

$${}_{p}L{\mathbb{A}}_{\theta +1,q}(\mu ,\eta ,\omega )=\left(\left(\mu +{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}\right){}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ),$$

(76)

which, in accordance with (27), we attain the assertion (70).

Again, by utilizing (73), by taking $f_q(\delta )=A_q(\delta )$ and $g_q(\delta )=e_q(\mu \delta )$, and following the same proof of (70), we obtain

$${}_{p}L{\mathbb{A}}_{\theta +1,q}(\mu ,\eta ,\omega )=\left(\left(\mu T_{\omega }+{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}\right){}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ),$$

(77)

which in view of (27), we attain the result (71).

On multiplying ${\widehat{D}}_{q,\mu }$ to the (62) and utilizing (11), we have

$$\delta A_q(\delta )e_q(\mu \delta ){\varphi }_q(\eta ,\delta )C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{\widehat{D}}_{q,\mu }{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(78)

which yields from (62). We also find the following:

$${\widehat{D}}_{q,\mu }\left\{{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )\right\}={\left[\theta \right]}_q{}_{p}L{\mathbb{A}}_{\theta -1,q}(\mu ,\eta ,\omega ),(\theta ⩾1),$$

(79)

which, in view of (28), yields assertion (72) of Theorem 7. □

Theorem 8. 

The following q-differential equation for ${}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ holds true:

$$\left(\mu {\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\widehat{D}}_{q,\mu }{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{T}_{\mu }\right.$$

$$+\left.{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{\widehat{D}}_{q,\mu }-{\left[\theta \right]}_q\right){}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=0,$$

(80)

and

$$\left(\mu {\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{T}_{\omega }+{\widehat{D}}_{q,\mu }{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}\right.$$

$$+\left.{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,x})}}{\widehat{D}}_{q,x}-{\left[\theta \right]}_q\right){}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=0.$$

(81)

Proof. 

By using Equations (70)–(72) in (30), we deduce

$$\left(\mu {\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\widehat{D}}_{q,\mu }{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{T}_{\mu }\right.$$

$$+\left.{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{\widehat{D}}_{q,\mu }\right){}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )={\left[\theta \right]}_q{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ),$$

(82)

and

$$\left(\mu {\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{T}_{\omega }+{\widehat{D}}_{q,\mu }{\displaystyle \frac{{\varphi }_q^{\prime }(\eta ,{\widehat{D}}_{q,\mu })}{{\varphi }_q(\eta ,{\widehat{D}}_{q,\mu })}}{T}_{\mu }{T}_{\omega }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}\right.$$

$$+\left.{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,x})}}{\widehat{D}}_{q,x}\right){}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )={\left[\theta \right]}_q{}_{p}L{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ).$$

(83)

Therefore, upon simplification, we obtain the assertions (80) and (81). □

3. Applications

This study extends the exploration of recently introduced polynomials, focusing on the examination of three-variable q-Laguerre polynomials. Now, substituting the value ${\varphi }_q(\eta ,\delta )=e_q(\eta {\delta }^2)$ in generating function (62), and then the results 3VqLP ${}_{p}L_{\theta ,q}(\mu ,\eta ,\omega )$ to the q-Hermite polynomials $H_{\theta ,q}(\mu ,\eta )$, it is demonstrated that the three-variable q-Laguerre–Hermite–Appell polynomials ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ can be characterized by a specific generating function.

$$A_q(\delta )e_q(\mu \delta )e_q(\eta {\delta }^2)C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(84)

In other words, we have

$${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=e_q(\eta {\widehat{D}}_{q,\mu }^2)e_q(-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu })\left\{A_{\theta ,q}(\mu )\right\}.$$

(85)

The Equation (84) can be transformed as follows:

$${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{A}_{\gamma ,q}{}_{L}H_{\theta -\gamma ,q}(\mu ,\eta ,\omega ).$$

(86)

Here, we provide the operational identities for $3VqLHAP$ ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$:

Theorem 9. 

The three-variable q-Laguerre–Hermite–Appell polynomials ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ satisfy the following respective operational identities:

$${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=e_q(-{\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu })\left\{{}_{H}A_{\omega ,q}(\mu ,\eta )\right\},$$

(87)

or, equivalently

$${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=C_{0,q}\left(\omega {\widehat{D}}_{q,\mu }\right)\left\{{}_{H}A_{\theta ,q}(\mu ,\eta )\right\},$$

(88)

or, equivalently

$${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=e_q(\eta {\widehat{D}}_{q,\mu }^2)\left\{{}_{L}A_{\theta ,q}(\mu ,\omega )\right\},$$

(89)

and

$$E_q\left({\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }\right){}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )={}_{H}A_{\theta ,q}(\mu ,\eta ),$$

(90)

or, equivalently

$$E_q\left(-\eta {\widehat{D}}_{q,\mu }^2\right){}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )={}_{L}A_{\theta ,q}(\mu ,\omega ).$$

(91)

Proof. 

In view of Equation (10), we have

$${\widehat{D}}_{q,\mu }^r{\mu }^n={\displaystyle \frac{{\left[\theta \right]}_q!}{{[\theta -r]}_q!}}{\mu }^{\theta -r}.$$

(92)

Using Equation (84), we obtain

$${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r{\omega }^r{\widehat{D}}_{q,\mu }^r}{{({\left[r\right]}_q!)}^2}}{}_{H}A_{\theta ,q}(\mu ,\eta ).$$

(93)

Utilizing (5) and (17) in (93), we attain the result of (87). Again utilizing (16) and (87), we acquire the result of (88). Furthermore, utilizing (17), and (20), we attain the assertion (89). On multiplying $E_q\left({\widehat{D}}_{q,\omega }^{-1}{\widehat{D}}_{q,\mu }\right)$ to the (87) and employing (8), we acquire the assertion (89). Again, multiplying $E_q\left(-\eta {\widehat{D}}_{q,\mu }^2\right)$ to the (89) and employing (8), we acquire the result (91). The proof of Theorem 9 is complete. □

Utilizing a similar approach as presented in [32,33] and considering Equation (84), we derive the following determinant form for ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ is obtained.

Theorem 10. 

The determinant representation of three-variable q-Laguerre–Hermite–Appell polynomials ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ of degree θ is

$${}_{L}H{\mathbb{A}}_{0,q}(\mu ,\eta ,\omega )={\displaystyle \frac{1}{{\beta }_{0,q}}},$$

$${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )={\displaystyle \frac{{(-1)}^{\theta }}{{({\beta }_{0,q})}^{\theta +1}}}\left|\begin{array}{cccccc}1& {}_{L}H_{1,q}(\mu ,\eta ,\omega )& {}_{L}H_{2,q}(\mu ,\eta ,\omega )& \dots & {}_{L}H_{\theta -1,q}(\mu ,\eta ,\omega )& {}_{L}H_{\theta ,q}(\mu ,\eta ,\omega )\\ {\beta }_{0,q}& {\beta }_{1,q}& {\beta }_{2,q}& \dots & {\beta }_{\theta -1,q}& {\beta }_{\theta ,q}\\ 0& {\beta }_{0,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\beta }_{1,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{1}}\right)}_q{\beta }_{\theta -2,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{1}}\right)}_q{\beta }_{\theta -1,q}\\ 0& 0& {\beta }_{0,q}& \dots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{1}}\right)}_q{\beta }_{\theta -3,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{2}}\right)}_q{\beta }_{\theta -2,q}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & {\beta }_{0,q}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\theta -1}}\right)}_q{\beta }_{1,q}\end{array}\right|,$$

(94)

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

where ${\beta }_{0,q}\ne 0,$ ${\beta }_{0,q}={\displaystyle \frac{1}{A_{0,q}}}$ and ${}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ),\theta =0,1,2,\dots ,$ are the three-variable q-Laguerre–Hermite polynomials.

Proof. 

By using the series manipulation method, the new generalization of the three-variable q-Laguerre–Hermite polynomials to the three-variable q-Laguerre–Hermite–Appell polynomials, we attain

$$A_q(\delta )\sum _{\theta =0}^{\infty }{}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}=\sum _{n=0}^{\infty }{}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(95)

Multiplying both sides by

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

(96)

we obtain

$$\sum _{\theta =0}^{\infty }{}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}=\sum _{\gamma =0}^{\infty }{\beta }_{\gamma ,q}{\displaystyle \frac{{\delta }^{\gamma }}{{\left[\gamma \right]}_q!}}\sum _{\theta =0}^{\infty }{}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(97)

Applying Cauchy product in (97) gives

$${}_{L}H_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{\beta }_{\gamma ,q}{}_{L}H{\mathbb{A}}_{\theta -\gamma ,q}(\mu ,\eta ,\omega ).$$

(98)

This equality leads to the system of $\theta$-equations with unknown ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega ),\theta =0,1,2,\cdots$.

To solve this system, we apply Cramer’s rule, utilizing the fact that the denominator is given by the determinant of a lower triangular matrix, which simplifies to ${\left({\beta }_{0,q}\right)}^{\theta +1}$. By transposing the numerator and systematically shifting the $i^{th}$ row by $(i+1)$-th; we obtain a well-structured solution. This approach ensures a precise and systematic determination of the unknowns while preserving the inherent properties of the system. Consequently, the desired result emerges in a mathematically elegant and computationally efficient manner. □

Next, we establish the q-multiplicative and q-derivative operators of ${}_{L}H{\mathbb{A}}_{n,q}(\zeta ,\eta ,\gamma )$, as formalized in the following theorem.

Theorem 11. 

The ${}_{L}H_{\theta ,q}(\mu ,\eta ,\omega )$ polynomials, known as three-variable q-Lagueree–Hermite–Appell polynomials exhibit quasi-monomials properties when subject to q-multiplicative and q-derivative operators:

$${\widehat{M}}_{3VqLHAP}=\left(\mu +\eta {\widehat{D}}_{q,\mu }{T}_{(\eta ;2)}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}},$$

(99)

or, equivalently,

$${\widehat{M}}_{3VqLHAP}=\left(\mu T_{\omega }+\eta {\widehat{D}}_{q,\mu }{T}_{(\eta ;2)}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}},$$

(100)

and

$${\widehat{P}}_{3VqLHAP}={\widehat{D}}_{q,\mu }.$$

(101)

Proof. 

Applying the partial q-derivative to the (84) with respect to $\delta$ and employing (15), we attain the following:

$$\sum _{\theta =1}^{\infty }{}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ){\widehat{D}}_{q,\delta }{\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}$$

$$=A_q(q\delta )e_q(q\mu \delta ){\widehat{D}}_{q,\theta }\left(e_q(y{\delta }^2)C_{0,q}(\omega \delta )\right)+e_q(\eta {\delta }^2)C_{0,q}(\omega \delta ){\widehat{D}}_{q,\delta }\left(A_q(\delta )e_q(\mu \delta )\right).$$

(102)

Using (15), (22), and (24) in (102), we acquire

$$\sum _{\theta =1}^{\infty }{}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta -1}}{{[\theta -1]}_q!}}$$

$$=\left(\left(\mu +\eta {\widehat{D}}_{q,\mu }{T}_{(\eta ;2)}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q\delta )}{A_q(\delta )}}+{\displaystyle \frac{A_q^{\prime }(\delta )}{A_q(\delta )}}\right)A_q(\delta )e_q(\mu \delta )e_q(\eta {\delta }^2)C_{0,q}(\omega \delta ).$$

(103)

Therefore, by using Equation (84), we obtain

$$\sum _{\theta =0}^{\infty }{}_{L}H_{\theta +1,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}$$

$$\sum _{n=0}^{\infty }\left(\left(\mu +\eta {\widehat{D}}_{q,\mu }{T}_{(\eta ;2)}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,x}\mu )}}\right){}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(104)

which yields the assertion (104). We also obtain

$${}_{L}H_{\theta +1,q}(\mu ,\eta ,\omega )=\left(\left(\mu +\eta {\widehat{D}}_{q,\mu }{T}_{(\eta ;2)}{T}_{\mu }{T}_{\omega }-{\widehat{D}}_{q,\omega }^{-1}{T}_{\mu }\right){\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}\right){}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ),$$

(105)

which, using Equation (27), yields assertion (99).

Again, using Equation (102), by taking $f_q(\delta )=e_q(\mu \delta )e_q(\eta {\delta }^2)$ and $g_q(\delta )=A_q(\delta )C_{0,q}(\omega \delta )$ and following the same process of assertion (99), we obtain the assertion (100).

Operating ${\widehat{D}}_{q,\mu }$ on both sides of Equation (84) by using Equation (11), we have

$$\delta A_q(\delta )e_q(\mu \delta )e_q(\eta {\delta }^2)C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{\widehat{D}}_{q,\mu }{}_{L}H_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(106)

which, in view of (84), yields the assertion; we find

$${\widehat{D}}_{q,\mu }\left\{{}_{L}H_{\theta ,q}(\mu ,\eta ,\omega )\right\}={\left[\theta \right]}_q{}_{L}H_{\theta -1,q}(\mu ,\eta ,\omega ),(\theta ⩾1),$$

(107)

which, in view of (28), yields the assertion (101) of Theorem 11. □

Theorem 12. 

The following q-differential equation for ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ is given by the following:

$$\left(\mu {\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}+\eta {\widehat{D}}_{q,\mu }^2{T}_{(\eta ;2)}{T}_{\mu }{T}_{\omega }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}-{\widehat{D}}_{q,\omega }^{-1}{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{\widehat{D}}_{q,\mu }{T}_{\mu }-{\left[\theta \right]}_q\right){}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=0,$$

(108)

and

$$\left(\mu {\widehat{D}}_{q,\mu }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{T}_{\omega }+\eta {\widehat{D}}_{q,\mu }^2{T}_{(\eta ;2)}{T}_{\mu }{T}_{\omega }{\displaystyle \frac{A_q(q{\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}-{\widehat{D}}_{q,\omega }^{-1}{\displaystyle \frac{A_q^{\prime }({\widehat{D}}_{q,\mu })}{A_q({\widehat{D}}_{q,\mu })}}{\widehat{D}}_{q,\mu }-{\left[\theta \right]}_q\right){}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )=0.$$

(109)

Proof. 

Using Equations (99)–(101) in (30), we obtain the assertions (108) and (109). □

4. Examples

In this section, we identify specific family members of the three-variable q-Laguerre–Hermite–Appell polynomials ${}_{L}H{\mathbb{A}}_{\theta ,q}(\mu ,\eta ,\omega )$ as defined in (84).

The following provides a suitable representation for specific members within the class of q-Appell polynomials:

The elements of the q-Appell polynomials class ${\mathcal{A}}_{\theta ,q}(\mu )$ corresponding q-numbers ${\mathcal{A}}_{\theta ,q}$ when evaluated at $\mu =0$. Table 1 illustrate the initial occurrences of three specific q-numbers: q-Bernoulli numbers ${\mathcal{B}}_{\theta ,q}$ [34,35], q-Euler numbers ${\mathcal{E}}_{\theta ,q}$ [34,35], and q-Genocchi numbers ${\mathcal{G}}_{\theta ,q}$ [34,35] (see Table 2).

Table 1. Some q-Appell polynomial families.

S. No.	q-Appell Polynomials	Generating Function	${\mathcal{A}}_{\mathit{q}}(\mathit{\delta })$
I.	The q-Bernoulli Polynomials [34,35]	${\displaystyle \frac{\delta }{e_q(\delta )-1}{e}_q(\mu \delta )}$  $={\displaystyle {\sum }_{\theta =0}^{\infty }}{\mathcal{B}}_{\theta ,q}(\mu ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}$	${\mathcal{A}}_q(\delta )={\displaystyle \frac{\delta }{e_q(\delta )-1}}$
II.	The q-Euler Polynomials [34,35]	${\displaystyle \frac{{\left[2\right]}_q}{e_q(\delta )+1}{e}_q(\mu \delta )}$  $={\displaystyle {\sum }_{\theta =0}^{\infty }}{\mathcal{E}}_{\theta ,q}(\mu ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}$	${\mathcal{A}}_q(\delta )={\displaystyle \frac{{\left[2\right]}_q}{e_q(\delta )+1}}$
III.	The q-Genocchi Polynomials [34,35]	${\displaystyle \frac{{\left[2\right]}_q\delta }{e_q(\delta )+1}{e}_q(\mu \delta )}$  $={\displaystyle {\sum }_{\theta =0}^{\infty }}{\mathcal{G}}_{\theta ,q}(\mu ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}$	${\mathcal{A}}_q(\delta )={\displaystyle \frac{{\left[2\right]}_q\delta }{e_q(\delta )+1}}$

Table 2. The first five q-numbers ${\mathcal{B}}_{\theta ,q}$, ${\mathcal{E}}_{\theta ,q}$ and ${\mathcal{G}}_{\theta ,q}$.

$\mathit{\theta }$	0	1	2	3	4
${\mathcal{B}}_{\theta ,q}$ [34,35]	1	${\displaystyle -{(1+q)}^{-1}}$	$q^2{({\left[3\right]}_q!)}^{-1}$	$(1-q)q^3{({\left[2\right]}_q)}^{-1}{({\left[4\right]}_q)}^{-1}$	$q^4(1-q^2-2q^3-q^4+q^6){({\left[2\right]}_q^2!{\left[3\right]}_q{\left[5\right]}_q)}^{-1}$
${\mathcal{E}}_{\theta ,q}$ [34,35]	1	${\displaystyle -\frac{1}{2}}$	${\displaystyle \frac{1}{4}}(-1+q)$	${\displaystyle \frac{1}{8}}(-1+2q+q^2-q^3)$	${\displaystyle \frac{1}{16}}(q-1){\left[3\right]}_q!(q^2-4q+1)$
${\mathcal{G}}_{\theta ,q}$ [34,35]	0	${\displaystyle \frac{2-q}{1+q}}$	${\displaystyle \frac{q(q-5)}{{(1+q)}^2}}$	$-{\displaystyle \frac{3q^2(q-5)}{{(1+q)}^3}}-{\displaystyle \frac{3q(2-q)}{{(1+q)}^2}}-{\displaystyle \frac{q}{(1+q)}}$	${\displaystyle \frac{-3q}{1+q}}\left({\displaystyle \frac{3q^3+10q^2-28q+7}{{(1+q)}^3}}\right)$

By carefully choosing the appropriate function $A_q(\delta )$ in Table 1, in Equation (84), it is possible to establish the following generating functions for the q-Laguerre–Hermite-based Bernoulli ${}_{L}H{\mathbb{B}}_{\theta ,q}(\mu ,\eta ,\omega )$, Euler ${}_{L}H{\mathbb{E}}_{\theta ,q}(\mu ,\eta ,\omega )$, and Genocchi ${}_{L}H{\mathbb{G}}_{\theta ,q}(\mu ,\eta ,\omega )$ polynomials:

$${\displaystyle \frac{\delta }{e_q(\delta )-1}}{e}_q(\mu \delta )e_q(\eta {\delta }^2)C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{}_{L}H{\mathbb{B}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(110)

$${\displaystyle \frac{2}{e_q(\delta )+1}}{e}_q(\mu \delta )e_q(\eta {\delta }^2)C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{}_{L}H{\mathbb{E}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}},$$

(111)

and

$${\displaystyle \frac{2\delta }{e_q(\delta )+1}}{e}_q(\mu \delta )e_q(\eta {\delta }^2)C_{0,q}(\omega \delta )=\sum _{\theta =0}^{\infty }{}_{L}H{\mathbb{G}}_{\theta ,q}(\mu ,\eta ,\omega ){\displaystyle \frac{{\delta }^{\theta }}{{\left[\theta \right]}_q!}}.$$

(112)

Further, in view of expression (86), the polynomials ${}_{L}H{\mathbb{B}}_{\theta ,q}(\mu ,\eta ,\omega )$, ${}_{L}H{\mathbb{E}}_{\theta ,q}(\mu ,\eta ,\omega )$, and ${}_{L}H{\mathbb{G}}_{\theta ,q}(\mu ,\eta ,\omega )$ satisfy the following explicit form:

$${}_{L}H{\mathbb{B}}_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{\mathbb{B}}_{\gamma ,q}{}_{L}H_{\theta -\gamma ,q}(\mu ,\eta ,\omega ),$$

(113)

$${}_{L}H{\mathbb{E}}_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{\mathbb{E}}_{\gamma ,q}{}_{L}H_{\theta -\gamma ,q}(\mu ,\eta ,\omega ),$$

(114)

and

$${}_{L}H{\mathbb{G}}_{\theta ,q}(\mu ,\eta ,\omega )=\sum _{\gamma =0}^{\theta }{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\gamma }}\right)}_q{\mathbb{G}}_{\gamma ,q}{}_{L}H_{\theta -\gamma ,q}(\mu ,\eta ,\omega ).$$

(115)

Furthermore, in view of expression (58) and by taking ${\beta }_{0,q}=1$, ${\beta }_{i,q}={\displaystyle \frac{1}{{[i+1]}_q}}(i=1,2,\cdots ,\theta )$, ${\beta }_{0,q}=1$, ${\beta }_{i,q}={\displaystyle \frac{1}{2}}(i=1,2,\cdots ,\theta )$ and ${\beta }_{0,q}=1$, ${\beta }_{i,q}={\displaystyle \frac{1}{2{[i+1]}_q}}(i=1,2,\cdots ,\theta )$ respectively, we find that the polynomials ${}_{L}H{\mathbb{B}}_{\theta ,q}(\mu ,\eta ,\omega )$, ${}_{L}H{\mathbb{E}}_{\theta ,q}(\mu ,\eta ,\omega )$ and ${}_{L}H{\mathbb{G}}_{\theta ,q}(\mu ,\eta ,\omega )$ satisfy the following determinant representations:

$${}_{L}H{\mathbb{B}}_{\theta ,q}(\mu ,\eta ,\omega )={(-1)}^{\theta }\left|\begin{array}{cccccc}1& {}_{L}H_{1,q}(\mu ,\eta ,\omega )& {}_{L}H_{2,q}(\mu ,\eta ,\omega )& \cdots & {}_{L}H_{\theta -1,q}(\mu ,\eta ,\omega )& {}_{L}H_{\theta ,q}(\mu ,\eta ,\omega )\\ & & & & & \\ 1& {\displaystyle \frac{1}{{\left[2\right]}_q}}& {\displaystyle \frac{1}{{\left[3\right]}_q}}& \cdots & {\displaystyle \frac{1}{{\left[\theta \right]}_q}}& {\displaystyle \frac{1}{{[\theta +1]}_q}}\\ & & & & & \\ 0& 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\displaystyle \frac{1}{{\left[2\right]}_q}}& \cdots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{1}}\right)}_q{\displaystyle \frac{1}{{[\theta -1]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{1}}\right)}_q{\displaystyle \frac{1}{{\left[\theta \right]}_q}}\\ & & & & & \\ 0& 0& 1& \cdots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{2}}\right)}_q{\displaystyle \frac{1}{{[\theta -2]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{2}}\right)}_q{\displaystyle \frac{1}{{[\theta -1]}_q}}\\ .& .& .& \cdots & .& \\ .& .& .& \cdots & .& \\ 0& 0& 0& \cdots & 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\theta -1}}\right)}_q{\displaystyle \frac{1}{{\left[2\right]}_q}}\end{array}\right|.$$

(116)

$${}_{L}H{\mathbb{E}}_{\theta ,q}(\mu ,\eta ,\omega )={(-1)}^{\theta }\left|\begin{array}{cccccc}1& {}_{L}H_{1,q}(\mu ,\eta ,\omega )& {}_{L}H_{2,q}(\mu ,\eta ,\omega )& \cdots & {}_{L}H_{\theta -1,q}(\mu ,\eta ,\omega )& {}_{L}H_{\theta ,q}(\mu ,\eta ,\omega )\\ & & & & & \\ 1& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& \cdots & {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\\ & & & & & \\ 0& 1& {\displaystyle \frac{1}{2}}{\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q& \cdots & {\displaystyle \frac{1}{2}}{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{1}}\right)}_q& {\displaystyle \frac{1}{2}}{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{1}}\right)}_q\\ & & & & & \\ 0& 0& 1& \cdots & {\displaystyle \frac{1}{2}}{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{2}}\right)}_q& {\displaystyle \frac{1}{2}}{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{2}}\right)}_q\\ .& .& .& \cdots & .& \\ .& .& .& \cdots & .& \\ 0& 0& 0& \cdots & 1& {\displaystyle \frac{1}{2}}{\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\theta -1}}\right)}_q\end{array}\right|$$

(117)

and

$${}_{L}H{\mathbb{E}}_{\theta ,q}(\mu ,\eta ,\omega )={(-1)}^{\theta }\left|\begin{array}{cccccc}1& {}_{L}H_{1,q}(\mu ,\eta ,\omega )& {}_{L}H_{2,q}(\mu ,\eta ,\omega )& \cdots & {}_{L}H_{\theta -1,q}(\mu ,\eta ,\omega )& {}_{L}H_{\theta ,q}(\mu ,\eta ,\omega )\\ & & & & & \\ 1& {\displaystyle \frac{1}{2{\left[2\right]}_q}}& {\displaystyle \frac{1}{2{\left[3\right]}_q}}& \cdots & {\displaystyle \frac{1}{2{\left[\theta \right]}_q}}& {\displaystyle \frac{1}{2{[\theta +1]}_q}}\\ \\ 0& 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right)}_q{\displaystyle \frac{1}{2{\left[2\right]}_q}}& \cdots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{1}}\right)}_q{\displaystyle \frac{1}{2{[\theta -1]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{1}}\right)}_q{\displaystyle \frac{1}{2{\left[\theta \right]}_q}}\\ & & & & & \\ \\ 0& 0& 1& \cdots & {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta -1}{2}}\right)}_q{\displaystyle \frac{1}{2{[\theta -2]}_q}}& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{2}}\right)}_q{\displaystyle \frac{1}{2{[\theta -1]}_q}}\\ .& .& .& \cdots & .& \\ .& .& .& \cdots & .& \\ 0& 0& 0& \cdots & 1& {\left({\displaystyle \genfrac{}{}{0pt}{}{\theta }{\theta -1}}\right)}_q{\displaystyle \frac{1}{2{\left[2\right]}_q}}\end{array}\right|.$$

(118)

5. Conclusions

The exploration and generalization of three-variable q-Laguerre–Appell polynomials represent a significant advancement in polynomial theory, with profound implications for quantum mechanics and entropy modeling. By utilizing the monomiality principle and operational techniques, these polynomials provide novel insights into previously unexplored mathematical domains. This study establishes precise formulas and highlights key properties, deepening our understanding of their structure and connections with other well-known polynomial families. Such developments not only expand the mathematical framework but also open avenues for future research. The potential applications of these polynomials extend across various disciplines, including quantum mechanics, mathematical physics, statistical mechanics, information theory, and computational science, making them a subject of considerable interest. Future investigations may focus on their structural intricacies and algebraic properties, uncovering deeper theoretical insights and practical implementations. Additionally, interdisciplinary collaborations could further enhance their real-world applicability, fostering innovation across multiple scientific fields.

In conclusion, the introduction and analysis of q-hybrid polynomials represent a significant breakthrough in mathematical research, offering new avenues for exploration across diverse scientific fields. Their rich structural properties and potential applications in areas such as quantum mechanics, statistical physics, and computational mathematics underscore their importance. Continued investigation, coupled with interdisciplinary collaboration, is essential to unlocking their full potential and deepening our understanding of their theoretical and practical implications. As research progresses, these polynomials may pave the way for novel mathematical frameworks and innovative solutions to complex scientific problems.