A New Generalization of mth-Order Laguerre-Based Appell Polynomials Associated with Two-Variable General Polynomials

Abstract

This paper presents a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized mth-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the mth-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations.

1. Introduction and Preliminary Results

It is well-established that special polynomials with two variables provide new analytical tools for solving a broad range of partial differential equations frequently encountered in physical problems. The introduction of the two-variable Laguerre polynomials, denoted as ${\mathcal{L}}_n(r_1,r_2)$ [1,2], is of significant interest due to its intrinsic mathematical properties and extensive applications in physics.

The higher-order Hermite or Gould–Hopper polynomials sometimes denoted by the Kampé de Fériet notation are defined as follows [3]:

$$g_n^m(r_1,r_2)={\mathcal{H}}_n^{\left(m\right)}(r_1,r_2)=n!\sum _{k=0}^{\left[{\displaystyle \frac{n}{m}}\right]}{\displaystyle \frac{r_2^k{r}_1^{n-mk}}{k!(n-mk)!}}.$$

(1)

The two-variable Hermite (or Gould–Hopper) polynomials (2VGHPs) ${\mathcal{H}}_n^{\left(m\right)}(r_1,r_2)$ are characterized by the following generating function [4,5,6]:

$$e^{r_{1}t+r_2{t}^m}=\sum _{n=0}^{\infty }{\mathcal{H}}_n^{\left(m\right)}(r_1,r_2){\displaystyle \frac{t^n}{n!}}.$$

(2)

The two-variable generalized Laguerre polynomials (2VGLPs) ${}_{\left[m\right]}{L}_n(r_1,r_2)$ are characterized by the following generating function [7]:

$$e^{r_{2}t}{C}_0(-r_1{t}^m)=e^{r_{2}t+{\widehat{D}}_{r_1}^{-1}{t}^m}=\sum _{n=0}^{\infty }{}_{\left[m\right]}{L}_n(r_1,r_2){\displaystyle \frac{t^n}{n!}},$$

(3)

where ${\widehat{D}}_{r_1}^{-1}$ is the inverse derivative operator of ${\widehat{D}}_{r_1}$ and $C_0\left(r_{1}t\right)$ represents the ordinary Bessel function of the first kind of order zero [4], defined as follows:

$$C_n\left(r_{1}t\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{r_1}^k}{k!(n+k)!}},n=0,1,2,\cdots ,.$$

(4)

Additionally, we note that

$${}_{\left[m\right]}{L}_n(r_1,r_2)=H_n^{\left(m\right)}\left(r_2,{\widehat{D}}_{r_1}^{-1}\right).$$

(5)

is the inverse differential operator.

The three-variable Laguerre–Gould–Hopper polynomials ${}_{\mathcal{L}}{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)$ are defined by the following [8]:

$$C_0(-r_1{t}^m)e^{r_{2}t+r_3{t}^s}=\sum _{n=0}^{\infty }{}_{\mathcal{L}}{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}},$$

(6)

The operational identities of generalized Laguerre–Gould–Hopper polynomials are

$${}_{\mathcal{L}}{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=exp\left(r_3{\displaystyle \frac{{\partial }^s}{{\partial }_{r_2^s}}}\right)\left\{{}_{\left[m\right]}{\mathcal{L}}_n(r_1,r_2)\right\},$$

and

$${}_{\mathcal{L}}{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=exp\left({\widehat{D}}_{r_1}^{-1}{\displaystyle \frac{{\partial }^m}{{\partial }_{r_2^m}}}\right)\left\{{\mathcal{H}}_n(r_2,r_3)\right\}.$$

The class of Appell polynomial sequences [9,10] appears in numerous applied mathematics problems, theoretical physics, approximation theory, and other mathematical disciplines. These sequences were defined through the following generating function:

$$\mathcal{R}(r_1,t):=\mathcal{R}\left(t\right)e^{r_{1}t}=\sum _{n=0}^{\infty }{\mathcal{R}}_n\left(r_1\right){\displaystyle \frac{t^n}{n!}},{\mathcal{R}}_n:={\mathcal{R}}_n\left(0\right),$$

(7)

where $\mathcal{R}\left(t\right)$ is an analytic function at $t=0$, expressed as

$$\mathcal{R}\left(t\right)=\sum _{n=0}^{\infty }{\mathcal{R}}_n{\displaystyle \frac{t^n}{n!}},{\mathcal{R}}_0\ne 0,{\mathcal{R}}_i(i=0,1,2,\cdots )\mathrm{being}\mathrm{real}\mathrm{coefficients}.$$

(8)

The Appell polynomials ${\mathcal{R}}_n\left(r_1\right)$ are explicitly given by the series expansion

$${\mathcal{R}}_n\left(r_1\right)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathcal{R}}_{n-k}{r}_1^k,{\mathcal{R}}_n^{\prime }\left(r_1\right)=n{\mathcal{R}}_{n-1}\left(r_1\right).$$

(9)

By appropriately selecting $\mathcal{R}\left(t\right)$, various members of the Appell polynomial family can be derived. These are listed in

Table 1. Certain members belonging to the Appell family.

S.No.	Name of Polynomials	$\mathcal{R}\left(\mathit{t}\right)$	Generating Function	Series Definition
I.	Bernoulli	${\displaystyle \frac{t}{e^t-1}}$	$\left({\displaystyle \frac{t}{e^t-1}}\right)e^{r_{1}t}={\displaystyle \sum _{n=0}^{\infty }}{\mathbb{B}}_n\left(r_1\right){\displaystyle \frac{t^n}{n!}}$	${\mathbb{B}}_n\left(r_1\right)={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathbb{B}}_k{r}_1^{n-k}$
	polynomials		$\left({\displaystyle \frac{t}{e^t-1}}\right)={\displaystyle \sum _{n=0}^{\infty }}{\mathbb{B}}_n{\displaystyle \frac{t^n}{n!}}$
	and numbers [7]		${\mathbb{B}}_n(:={\mathbb{B}}_n\left(0\right)={\mathbb{B}}_n\left(1\right))$
II.	Euler	${\displaystyle \frac{2}{e^t+1}}$	$\left({\displaystyle \frac{2}{e^t+1}}\right)e^{r_{1}t}={\displaystyle \sum _{n=0}^{\infty }}{\mathbb{E}}_n\left(r_1\right){\displaystyle \frac{t^n}{n!}}$	${\mathbb{E}}_n\left(r_1\right)={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{{\mathbb{E}}_k}{2^k}}{\left(r_1-{\displaystyle \frac{1}{2}}\right)}^{n-k}$
	polynomials		${\displaystyle \frac{2e^t}{e^{2t}+1}}={\displaystyle \sum _{n=0}^{\infty }}{\mathbb{E}}_n{\displaystyle \frac{t^n}{n!}}$
	and numbers [7]		$E_n:=2^n{\mathbb{E}}_n\left({\displaystyle \frac{1}{2}}\right)$
III.	Genocchi	${\displaystyle \frac{2t}{e^t+1}}$	$\left({\displaystyle \frac{2t}{e^t+1}}\right)e^{r_{1}t}={\displaystyle \sum _{n=0}^{\infty }}{\mathbb{G}}_n\left(r_1\right){\displaystyle \frac{t^n}{n!}}$	${\mathbb{G}}_n\left(r_1\right)={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathbb{G}}_k{r}_1^{n-k}$
	polynomials		${\displaystyle \frac{2t}{e^t+1}}={\displaystyle \sum _{n=1}^{\infty }}{\mathbb{G}}_n{\displaystyle \frac{t^n}{n!}}$
	and numbers [7]		${\mathbb{G}}_n:={\mathbb{G}}_n\left(0\right)$

below:

To facilitate further computations, we present the initial values of Bernoulli numbers ${\mathbb{B}}_n$, Euler numbers ${\mathbb{E}}_n$, and Genocchi numbers ${\mathbb{G}}_n$ in

Table 2. Values for four degrees of ${\mathbb{B}}_n$, ${\mathbb{E}}_n$, and ${\mathbb{G}}_n$.

n	0	1	2	3	4
${\mathbb{B}}_n$	1	$\pm {\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{6}}$	0	$-{\displaystyle \frac{1}{30}}$
${\mathbb{E}}_n$	1	0	−1	0	5
${\mathbb{G}}_n$	0	1	−1	0	1

Note: It is evident from the above table that the polynomial sequence ${\mathbb{G}}_n\left(r_1\right)$ has degree $n-1$, unlike the other Appell polynomials, which all have degree n. Hence, ${\mathbb{G}}_n\left(r_1\right)$ does not belong to the class of strongly Appell polynomial sequences (see [11,12] for details).

below:

In [13], Özat et al. introduced and analyzed a hybrid class of truncated-exponential-based general-Appell polynomials, denoted as ${}_{e^{\left(m\right)}}{\mathcal{T}}_n(r_1,r_2,r_3)$, which are defined via the following generating function:

$$\mathcal{R}\left(t\right)exp\left(r_{1}t\right)\mathrm{\Psi }(r_2,t){\displaystyle \frac{1}{1-r_3{t}^m}}=\sum _{n=0}^{\infty }{}_{e^{\left(m\right)}}{\mathcal{T}}_n(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}},m\in \mathbb{N},\mid r_{3}t\mid <1.$$

(10)

The operational identities of three-variable generalized truncated-exponential-based general-Appell polynomials are given as

$${}_{e^{\left(m\right)}}{\mathcal{T}}_n(r_1,r_2,r_3)=exp\left(r_3{\widehat{D}}_{r_3}{r}_3{\widehat{D}}_{r_1}^m\right)\left\{{}_{\mathcal{P}}{\mathcal{R}}_n(r_1,r_2)\right\}.$$

(11)

The two-variable general polynomials (2VgPs) denoted by ${\mathcal{P}}_n(r_1,r_2)$ are specified by the following generating relation [14]:

$$exp\left(r_{1}t\right)\mathrm{\Psi }(r_2,t)=\sum _{n=0}^{\infty }{\mathcal{P}}_n(r_1,r_2){\displaystyle \frac{t^n}{n!}},({\mathcal{P}}_0(r_1,r_2)=1),$$

(12)

where $\mathrm{\Psi }(r_2,t)$ has (at least the formal) series expansion

$$\mathrm{\Psi }(r_2,t)=\sum _{k=0}^{\infty }{\mathrm{\Psi }}_k\left(r_2\right){\displaystyle \frac{t^k}{k!}},({\mathrm{\Psi }}_0\left(r_2\right)\ne 0).$$

(13)

The foundational idea of the monomiality principle dates back to 1941 when Steffenson [15] first introduced the concept through the notion of poweroid. This approach was later refined and extended by Dattoli [16,17], paving the way for further advancements in the field.

The monomiality principle states that the operators $\widehat{\mathcal{M}}$ and $\widehat{\mathcal{P}}$ act as multiplicative and differential operators, respectively, for a given polynomial sequence ${\left\{q_n\left(r_1\right)\right\}}_{n\in \mathbb{N}}$. More specifically, these operators satisfy the fundamental recurrence relations

$$q_{n+1}\left(r_1\right)=\widehat{\mathcal{M}}\left\{q_n\left(r_1\right)\right\},$$

(14)

and

$$n{q}_{n-1}\left(r_1\right)=\widehat{\mathcal{P}}\left\{q_n\left(r_1\right)\right\}.$$

(15)

A polynomial sequence ${\left\{q_n\left(r_1\right)\right\}}_{n\in \mathbb{N}}$ that adheres to these operator relations is referred to as a quasi-monomial set. Such a set must also satisfy the fundamental commutation relation

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

(16)

which aligns naturally with the algebraic framework of the Weyl algebra.

If a polynomial sequence ${\left\{q_n\left(r_1\right)\right\}}_{n\in \mathbb{N}}$ is quasi-monomial, its defining properties can be derived directly from the characteristics of the operators $\widehat{\mathcal{M}}$ and $\widehat{\mathcal{P}}$. Specifically, the following key properties were established:

(i)

The polynomials $q_n\left(r_1\right)$ satisfy a differential equation of the form

$$\widehat{\mathcal{M}}\widehat{\mathcal{P}}\left\{q_n\left(r_1\right)\right\}=n{q}_n\left(r_1\right),$$

(17)

provided that $\widehat{\mathcal{M}}$ and $\widehat{\mathcal{P}}$ admit suitable differential representations.

(ii)

An explicit formula for $q_n\left(r_1\right)$ can be expressed as

$$q_n\left(r_1\right)={\widehat{\mathcal{M}}}^n\left\{1\right\},$$

(18)

with the initial condition $q_0\left(r_1\right)=1$.

(iii)

The exponential generating function of $q_n\left(r_1\right)$ is given by

$$e^{t\widehat{\mathcal{M}}}\left\{1\right\}=\sum _{n=0}^{\infty }{q}_n\left(r_1\right){\displaystyle \frac{t^n}{n!}}\left(\left|t\right|<\infty \right),$$

(19)

which follows directly from Equation (19).

The operational framework outlined above has been applied extensively applications across various fields, including classical optics, quantum mechanics, and different branches of mathematical physics. These techniques offer robust analytical tools for studying diverse polynomial families.

Inspired by recent advancements, this paper introduces a novel generalization of the mth-order Laguerre-based Appell polynomials, denoted by ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$. This new class of polynomials extends the classical Laguerre and Appell frameworks by incorporating multi-parameter structures and fractional operators, leading to a richer algebraic and operational theory. Section 2 formulates these generalized polynomials and investigates their foundational properties, including recurrence relations, multiplicative and derivative operators, and associated differential equations. Section 3 presents both series expansions and determinantal representations, offering explicit closed-form insights. In Section 4, we identify and analyze several subfamilies, such as the generalized mth-order Laguerre–Hermite Appell polynomials, which unify and extend known families. Section 5 presents a further generalization via fractional calculus, leading to the development of mth-order Laguerre–Gould–Hopper-based Appell polynomials. Additionally, novel hybrid families namely Laguerre–Gould–Hopper–Bernoulli, Laguerre–Gould–Hopper–Euler, and Laguerre–Gould–Hopper–Genocchi polynomials are introduced and studied. The original contributions of this work lie in the systematic construction of these new polynomial families, the derivation of their structural and operational properties using the quasi-monomiality principle and determinant techniques, and the unification of various classical polynomials under a broader generalized framework. These advancements offer new tools for applications in mathematical physics, special function theory, and the study of fractional differential equations.

2. The New Generalization of $\mathit{m}$th-Order Laguerre and Laguerre-Based Appell Polynomials

In this part of the paper, we develop a novel extension of the three-variable mth-order Laguerre-based Appell polynomials, represented by ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$. We explore their formal series representation, analyze their quasi-monomial structure, and derive associated operational identities and differential equations. The discussion begins with the formulation of a new class of generalized mth-order three-variable Laguerre polynomials, denoted by 3VLP ${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)$.

Utilizing the relations (3) and (12), we introduce the new generalization of mth-order three variable Laguerre polynomials ${}_{\mathcal{P}}{\mathcal{L}}_{\upsilon }(r_1,r_2,r_3)$ in the following form:

$$e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)=\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}},({\mathcal{P}}_0(r_1,r_2)=1).$$

(20)

Utilizing Equations (3) and (13) to simplify the left-hand side of Equation (20), we obtain the following series expansions for the generalized mth-order three-variable Laguerre polynomials ($3mVLP$), denoted by ${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)$:

$${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right){\mathrm{\Psi }}_k\left(r_2\right){}_{\left[m\right]}{\mathcal{L}}_{n-k}(r_1,r_3).$$

(21)

We present the derived quasi-monomial identities for the mth-order three-variable Laguerre polynomials ($m3VLP$) denoted as ${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)$.

Theorem 1. 

The new generalization of ${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)$ demonstrates quasi-monomial properties under the following multiplicative and derivative operators:

$${\widehat{M}}_{3VgLP}=r_1+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1},$$

(22)

and

$${\widehat{P}}_{3VgLP}={\widehat{D}}_{r_1},$$

(23)

where ${\widehat{D}}_{r_3}^{-1}$ is the inverse derivative operator of ${\widehat{D}}_{r_3}$.

Proof. 

By differentiating Equation (20) partially with respect to t on both sides, it follows that

$$\left(r_1+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,t)}{\mathrm{\Psi }(r_2,t)}}+m{D}_{r_3}^{-1}{t}^{m-1}\right)\left\{e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)\right\}=\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(24)

If $\mathrm{\Psi }(r_2,t)$ is an invertible series and ${\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,t)}{\mathrm{\Psi }(r_2,t)}}$ has Taylor’s series expansion in powers of t, then in view of the identity

$${\widehat{D}}_{r_1}\left\{e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)\right\}=t{e}^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m),$$

(25)

we can write

$${\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}\left\{e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)\right\}={\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,t)}{\mathrm{\Psi }(r_2,t)}}\left\{e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)\right\}.$$

(26)

Now, using (26) in the l.h.s of (24), we have

$$\left(r_1+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}+m{D}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\right)\left\{e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)\right\}=\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(27)

Using Equation (20) in the l.h.s. of the above equation, we have

$$\sum _{n=0}^{\infty }\left(r_1+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}+m{D}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\right){}_{\mathcal{P}}{\mathcal{L}}_n^{(m+1)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(28)

In light of (14) and (28), we obtain the assertion (22).

Similarly, by applying identity (25) to (20), we obtain

$${\widehat{D}}_{r_1}\left\{\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right\}=\sum _{n=1}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{(n-1)!}}$$

By matching the coefficients of the same exponents of t on both sides of the above equation, it follows that

$${\widehat{D}}_{r_1}\left\{{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)\right\}=n{}_{\mathcal{P}}{\mathcal{L}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3),n⩾1.$$

(29)

Thus in view of (15) and (29), we obtain Assertion (23). □

Theorem 2. 

The differential equations for mth-order three-variable generalized Laguerre polynomials ($3VGLP$), ${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)$, are given as

$$\left(r_1{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}{\widehat{D}}_{r_1}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^m-n\right){}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)=0,$$

(30)

Proof. 

In view of Equations (22) and (23) in (17), we have

$$\left(r_1{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}{\widehat{D}}_{r_1}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\right){}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)=n{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3).$$

Upon solving the above equation, we obtain Assertion (30) from Theorem 2.2. □

Remark 1. 

Since ${\mathcal{P}}_0(r_1,r_2)=1$, in view of the monomiality principle given in Equation (18), we have

$${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)={\left(r_1+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\right)}^n\left\{1\right\},({\mathcal{P}}_0(r_1,r_2)=1).$$

Also, in view of Equations (19), (20) and (22), we have

$$exp\left({\widehat{M}}_{3VgP}\right)\left\{1\right\}=e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)=\sum _{\upsilon =0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

We now turn our attention to formulating a novel generalization of the mth-order three-variable Laguerre-based Appell polynomials (G3VLbAP). To construct their generating function, we make use of the exponential generating function typically employed for Appell-type polynomials. Specifically, by substituting $r_1$ on the left-hand side of Equation (7) with the multiplicative operator ${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)$ defined in (22), which represents the newly generalized mth-order three-variable Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$, we obtain

$$\mathcal{R}\left(t\right)exp\left({\widehat{M}}_{3VgP}\right)\left\{1\right\}=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}},$$

(31)

by which, on using Equation (22), we obtain the following two equivalent forms of ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$:

$$\mathcal{R}\left(t\right)exp\left(r_1+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\right)\left\{1\right\}=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(32)

Using Relation (31) on the left-hand side of Equation (32), the generating function for the new generalization of mth-order three-variable Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$ has the following form:

$$\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(33)

where

$$\mathcal{R}\left(t\right)=\sum _{k=0}^{\infty }{\alpha }_k{\displaystyle \frac{t^k}{k!}},{\alpha }_0\ne 0\mathrm{\Psi }(r_2,t)=\sum _{k=0}^{\infty }{\psi }_k\left(r_2\right){\displaystyle \frac{t^k}{k!}},{\psi }_0\ne 0.$$

(34)

Theorem 3. 

The generalized mth-order Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$ satisfy the following recurrence relation:

$${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-k}^{\left(m\right)}(r_1,r_2,r_3){\gamma }_k+r_1{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)+\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-k}^{\left(m\right)}(r_1,r_2,r_3){\rho }_k\left(r_2\right)$$

$$+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3).$$

(35)

where

$${\displaystyle \frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}}=\sum _{k=0}^{\infty }{\gamma }_k{\displaystyle \frac{t^k}{k!}},{\displaystyle \frac{{\mathrm{\Psi }}_t(r_2,t)}{\mathrm{\Psi }(r_2,t)}}=\sum _{k=0}^{\infty }{\rho }_k\left(r_2\right){\displaystyle \frac{t^k}{k!}},{\mathrm{\Psi }}_t(r_2,t)={\displaystyle \frac{\partial }{\partial t}}\mathrm{\Psi }(r_2,t)$$

(36)

and ${\widehat{D}}_{r_3}^{-1}$ is the inverse of ${\widehat{D}}_{r_3}$.

Proof. 

By differentiating both sides of Equation (33) with respect to t, we obtain the following:

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}}\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)+r_1\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)$$

$$+{\displaystyle \frac{{\mathrm{\Psi }}_t(r_2,t)}{\mathrm{\Psi }(r_2,t)}}\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)+m{\widehat{D}}_{r_3}^{-1}{t}^{m-1}\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)$$

(37)

Using (36), we have

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\left(\sum _{k=0}^{\infty }{\gamma }_k{\displaystyle \frac{t^k}{k!}}\right)\left(\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right)+\left(\sum _{k=0}^{\infty }{\rho }_k\left(r_2\right){\displaystyle \frac{t^k}{k!}}\right)\left(\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right)$$

$$+r_1\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(38)

Hence, when use the Cauchy product, we obtain

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-k}^{\left(m\right)}(r_1,r_2,r_3){\gamma }_k{\displaystyle \frac{t^n}{n!}}+\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\rho }_k\left(r_2\right){}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-k}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}$$

$$+r_1\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(39)

Thus, equating the coefficients of ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above Equation (39), we obtain Assertion (35). □

Theorem 4. 

The generalized mth-order Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$ satisfy the multiplicative and derivative operators as follows:

$$\widehat{M}=r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1},$$

(40)

and

$$\widehat{P}={\widehat{D}}_{r_1},$$

(41)

respectively.

Proof. 

Taking the derivative with respect to t on both sides of (33), we have

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}}\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)+r_1\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)$$

$$+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,t)}{\mathrm{\Psi }(r_2,t)}}\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)+m{D}_{r_3}^{-1}{t}^{m-1}\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)$$

(42)

$$\begin{array}{c}\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\left(r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}}+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,t)}{\mathrm{\Psi }(r_2,t)}}\right)\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)\hfill \\ \hfill +m{\widehat{D}}_{r_3}^{-1}{t}^{m-1}\mathcal{R}\left(t\right)e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m).\end{array}$$

(43)

By using Equation (33), we obtain

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n+1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left(r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\right){}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(44)

In view of (14) and (44), we have Assertion (40).

Again, partially differentiating Equation (33) w.r.to $r_1$ on both sides, it follows that

$${\widehat{D}}_{r_1}\left\{\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right\}=\sum _{n=1}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{(n-1)!}}.$$

(45)

By matching the coefficients of same exponents of t on both sides of (45), it follows that

$${\widehat{D}}_{r_1}\left\{{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)\right\}=n{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3),n⩾1.$$

(46)

Thus, in view of (15) and (46), we obtain Assertion (41). □

Theorem 5. 

The generalized mth-order Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$ satisfy the differential equation as follows:

$$\left(r_1{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathrm{\Psi }}^{\prime }(r_2,{\widehat{D}}_{r_1})}{\mathrm{\Psi }(r_2,{\widehat{D}}_{r_1})}}{\widehat{D}}_{r_1}+m{\widehat{D}}_{r_1}^m{\widehat{D}}_{r_3}^{-1}-n\right){}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)=0.$$

(47)

Proof. 

In view of Equations (40) and (41) in (17), we obtain Assertion (47). So, we omit the proof. □

3. Series Representation, Determinant Form, and Operational Identities

Hybrid special polynomials play a crucial role in mathematical analysis due to their rich structural properties. Their series representation provides explicit forms and recurrence relations, aiding in solving differential and functional equations. The determinant form of these polynomials offers a compact and elegant way to analyze their algebraic and combinatorial properties. It facilitates the study of orthogonality, symmetry, and transformation identities. Hybrid polynomials also bridge classical and modern polynomial families, extending their applicability in mathematical physics and engineering. Their determinant representation helps in computing higher-order coefficients efficiently. These polynomials are widely used in approximation theory and numerical analysis. Overall, they contribute significantly to both theoretical and applied mathematical research.

Theorem 6. 

The mth-order three-variable generalized Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$ are defined by the following series:

$${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathcal{R}}_k{}_{\mathcal{P}}{\mathcal{L}}_{n-k}^{\left(m\right)}(r_1,r_2,r_3),$$

(48)

with ${\mathcal{R}}_k$ given in Equation (8).

Proof. 

In view of Equation (33), we can write

$$\sum _{\upsilon =0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(49)

Using the expansion in (8) of $\mathcal{R}\left(t\right)$ from the left-hand side of Equation (49), we simplify and then equate the coefficients of like powers of $\delta$ on both sides of the resulting equation, leading us to Assertion (48). □

Theorem 7. 

The generalized mth-order Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$ have the following determinant representation:

$${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n,q}^{\left(m\right)}(r_1,r_2,r_3)={\displaystyle \frac{{(-1)}^n}{{\left({\beta }_0\right)}^{n+1}}}\left|\begin{array}{cccccc}1& {}_{\mathcal{P}}{\mathcal{L}}_1^{\left(m\right)}(r_1,r_2,r_3)& {}_{\mathcal{P}}{\mathcal{L}}_2^{\left(m\right)}(r_1,r_2,r_3)& \dots & {}_{\mathcal{P}}{\mathcal{L}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3)& {}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)\\ {\beta }_0& {\beta }_1& {\beta }_2& \dots & {\beta }_{n-1}& {\beta }_n\\ 0& {\beta }_0& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\beta }_1& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-2}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\beta }_{n-1}\\ 0& 0& {\beta }_0& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-3}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right){\beta }_{n-2}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & {\beta }_0& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\beta }_1\end{array}\right|,$$

(50)

where ${\beta }_0,{\beta }_1,{\beta }_2,\dots ,{\beta }_n$ are the coefficients of the Maclaurin series of the function ${\displaystyle \frac{1}{\mathcal{R}\left(t\right)}}$, and ${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)$ gives the new generalization of mth-order three-variable Laguerre polynomials in Equation (20).

Proof. 

Using the series representation of ${\displaystyle \frac{1}{\mathcal{R}\left(t\right)}}$

$${\left[\mathcal{R}\left(t\right)\right]}^{-1}=\sum _{k=0}^{\infty }{\beta }_k{\displaystyle \frac{t^k}{k!}},$$

and the generation function in (20), we obtain

$$e^{r_{1}t}\mathrm{\Psi }(r_2,t)C_0(-r_3{t}^m)=\left(\sum _{k=0}^{\infty }{\beta }_k{\displaystyle \frac{t^k}{k!}}\right)\left(\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right).$$

Hence

$$\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\left(\sum _{k=0}^{\infty }{\beta }_k{\displaystyle \frac{t^k}{k!}}\right)\left(\sum _{n=0}^{\infty }{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right).$$

Applying the Cauchy product, we have

$$\sum _{n=0}^{\infty }{}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\beta }_k{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-k}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

By comparing the coefficients of ${\displaystyle \frac{t^n}{n!}}$ from the polynomial equation, we obtain

$${}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\beta }_k{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-k}_{\mathrm{}}^{\mathrm{}}(r_1,r_2,r_3),n\in {\mathbb{N}}_0.$$

So, we obtain the system of equations as follows:

$${}_{\mathcal{P}}{\mathcal{L}}_0^{\left(m\right)}(r_1,r_2,r_3)={\beta }_0{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_0^{\left(m\right)}(r_1,r_2,r_3),$$

$${}_{\mathcal{P}}{\mathcal{L}}_1^{\left(m\right)}(r_1,r_2,r_3)={\beta }_0{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_1^{\left(m\right)}(r_1,r_2,r_3)+{\beta }_1{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_0^{\left(m\right)}(r_1,r_2,r_3),$$

$${}_{\mathcal{P}}{\mathcal{L}}_2^{\left(m\right)}(r_1,r_2,r_3)={\beta }_0{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_2^{\left(m\right)}(r_1,r_2,r_3)+\left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\beta }_1{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_1^{\left(m\right)}(r_1,r_2,r_3)+{\beta }_2{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_0^{\left(m\right)}(r_1,r_2,r_3),$$

$$⋮$$

$${}_{\mathcal{P}}{\mathcal{L}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3)={\beta }_0{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3)+\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_1{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-2}^{\left(m\right)}(r_1,r_2,r_3)+\dots +{\beta }_{n-1}{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_0^{\left(m\right)}(r_1,r_2,r_3),$$

$${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)={\beta }_0{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)+\left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\beta }_1{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3)+\dots +{\beta }_n{}_{\mathcal{P}\mathcal{L}}{\mathcal{R}}_0^{\left(m\right)}(r_1,r_2,r_3).$$

Applying Cramer’s rule, we get

$${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)={\displaystyle \frac{\left|\begin{array}{ccccc}{\beta }_0& 0& \dots & 0& {}_{\mathcal{P}}{\mathcal{L}}_0^{\left(m\right)}(r_1,r_2,r_3)\\ {\beta }_1& {\beta }_0& \dots & 0& {}_{\mathcal{P}}{\mathcal{L}}_1^{\left(m\right)}(r_1,r_2,r_3)\\ {\beta }_2& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\beta }_1& \dots & 0& {}_{\mathcal{P}}{\mathcal{L}}_2^{\left(m\right)}(r_1,r_2,r_3)\\ {\beta }_3& \left(\genfrac{}{}{0pt}{}{3}{2}\right){\beta }_2& \dots & 0& {}_{\mathcal{P}}{\mathcal{L}}_3^{\left(m\right)}(r_1,r_2,r_3)\\ ⋮& ⋮& ⋮& ⋮& ⋮& \\ {\beta }_{n-1}& \left(\genfrac{}{}{0pt}{}{n-1}{1}\right){\beta }_{n-2}& \dots & {\beta }_0& {}_{\mathcal{P}}{\mathcal{L}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3)\\ {\beta }_n& \left(\genfrac{}{}{0pt}{}{n}{1}\right){\beta }_{n-1}& \dots & \left(\genfrac{}{}{0pt}{}{n}{n-1}\right){\beta }_1& {}_{\mathcal{P}}{\mathcal{L}}_r^{\left(m\right)}(r_1,r_2,r_3)\end{array}\right|}{\left|\begin{array}{ccccc}{\beta }_0& 0& \dots & 0& 0\\ {\beta }_1& {\beta }_0& \dots & 0& 0\\ {\beta }_2& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\beta }_1& \dots & 0& 0\\ {\beta }_3& \left({\displaystyle \genfrac{}{}{0pt}{}{3}{2}}\right){\beta }_2& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& \\ {\beta }_{n-1}& \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-2}& \dots & {\beta }_0& 0\\ {\beta }_n& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\beta }_{n-1}& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\beta }_1& {\beta }_0\end{array}\right|}}.$$

By taking the transpose in the last equation, we have

$${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)={\displaystyle \frac{1}{{\left({\beta }_0\right)}^{n+1}}}\left|\begin{array}{ccccc}{\beta }_0& {\beta }_1& \dots & {\beta }_{n-1}& {\beta }_n\\ 0& {\beta }_0& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-2}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\beta }_{n-1}\\ 0& 0& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-3}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right){\beta }_{n-2}\\ ⋮& ⋮& ⋮& ⋮& ⋮& \\ 0& 0& \dots & {\beta }_0& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\beta }_1\\ {}_{\mathcal{P}}{\mathcal{L}}_0^{\left(m\right)}(r_1,r_2,r_3)& {}_{\mathcal{P}}{\mathcal{L}}_1^{\left(m\right)}(r_1,r_2,r_3)& \dots & {}_{\mathcal{P}}{\mathcal{L}}_{n-1}^{\left(m\right)}(r_1,r_2,r_3)& {}_{\mathcal{P}}{\mathcal{L}}_n^{\left(m\right)}(r_1,r_2,r_3)\end{array}\right|.$$

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

Theorem 8. 

The following operational representation connecting the mth-order three-variable generalized Laguerre-based Appell polynomials ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)$ holds true:

$${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)=exp\left(D_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{{\partial }_{r_1^m}}}\right)\left\{{}_{\mathcal{P}}{\mathcal{R}}_n(r_1,r_2)\right\}$$

(51)

Proof. 

By (33), we have

$${\displaystyle \frac{{\partial }^m}{{\partial }_{r_1^m}}}{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3)={\displaystyle \frac{\partial }{\partial D_{r_3}^{-1}}}{}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}(r_1,r_2,r_3).$$

(52)

Since, in view of (10), we have

$${}_L{R}_n^{(m,r)}(r_1,r_2,0)={}_L{R}_n^{\left(m\right)}(r_1,r_2).$$

(53)

Now, solving Equations (52) subject to initial condition (53), we obtain Assertion (51). □

4. Applications

This work advances the study of newly developed polynomial classes by focusing on a generalization of three-variable Laguerre-based Appell polynomials. In particular, by selecting the function $\mathrm{\Psi }(r_2,t)=e^{r_2{t}^s}$ within the generating function framework in (33), the generalized family ${}_{{}_{\mathcal{P}}\mathcal{L}}{\mathcal{R}}_n^{\left(m\right)}_{\mathrm{}}^{\mathrm{}}(r_1,r_2,r_3)$ simplifies to the Laguerre–Gould–Hopper-based Appell polynomials (LGHbAP), denoted as ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}_{\mathrm{}}^{\mathrm{}}(r_1,r_2,r_3)$, which are governed by a distinct generating function structure:

$$\mathcal{R}\left(t\right)e^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}_{\mathrm{}}^{\mathrm{}}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(54)

From some operational representations of Laguerre–Gould–Hopper-based Appell polynomials $\left(LGHbAP\right)$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{\left(m\right)}_{\mathrm{}}^{\mathrm{}}(r_1,r_2,r_3)$, it follows that

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)=exp\left({\widehat{D}}_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}+r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)\left\{{\mathcal{R}}_n\left(r_1\right)\right\},$$

(55)

and

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)=exp\left({\widehat{D}}_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}\right)\left\{{}_H{\mathcal{R}}_n^{\left(s\right)}(r_1,r_2)\right\},$$

or, equivalently,

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)=exp\left(r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)\left\{{}_{{}_{\left[m\right]}\mathcal{L}}{\mathcal{R}}_n(r_1,r_3)\right\}.$$

Theorem 9. 

The mth-order three-variable Laguerre–Gould–Hopper-based Appell polynomials are defined by the following series:

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}_{\mathrm{}}^{\mathrm{}}(r_1,r_2,r_3)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathcal{R}}_k{}_{{}_{\mathcal{L}}\mathcal{H}}_{n-k}^{(m,s)}(r_1,r_2,r_3).$$

(56)

Proof. 

In view of (54), we have

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(57)

Now, by using the expansion of $\mathcal{R}\left(t\right)$ (8) from the left-hand side of Equation (57), we can simplify and equate the coefficients of like powers of t on both sides of the resulting equation to obtain Assertion (56). □

The determinant form serves as a powerful and elegant tool in the study of special polynomials, providing a concise representation that encapsulates their fundamental properties. This structured formulation not only highlights key features such as orthogonality, recurrence relations, and generating functions but also facilitates their analytical and algebraic manipulation across a wide range of mathematical disciplines. By bridging connections with linear algebra and combinatorial identities, the determinant form deepens our understanding of the intrinsic structure and symmetries inherent in special polynomials. Beyond its computational utility, this approach contributes profoundly to theoretical advancements, offering fresh perspectives and unifying principles in the broader context of special functions and orthogonal polynomial theory.

Costabile and Longo [18], building upon the methodologies developed in [19,20], successfully derived the determinantal representation of Appell polynomials, offering a compact and structured framework that highlights their algebraic and analytical properties.

Theorem 10. 

The determinant representation of mth-order three-variable Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ of degree n is

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_0}^{(m,s)}(r_1,r_2,r_3)={\displaystyle \frac{1}{{\beta }_0}},$$

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)={\displaystyle \frac{{(-1)}^n}{{\left({\beta }_0\right)}^{n+1}}}\left|\begin{array}{cccccc}1& {}_{{}_{\mathcal{L}}\mathcal{H}}_1^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_2(r_1,r_2,r_3)& \dots & {}_{{}_{\mathcal{L}}\mathcal{H}}_{n-1}^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)\\ {\beta }_0& {\beta }_1& {\beta }_2& \dots & {\beta }_{n-1}& {\beta }_n\\ 0& {\beta }_0& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\beta }_1& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-2}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\beta }_{n-1}\\ 0& 0& {\beta }_0& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-3}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right){\beta }_{n-2}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & {\beta }_{0,q}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\beta }_1\end{array}\right|,$$

(58)

$${\beta }_n=-{\displaystyle \frac{1}{{\mathcal{R}}_0}}\left(\sum _{k=1}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathcal{R}}_k{\beta }_{n-k}\right),n=0,1,2,\dots ,$$

where ${\beta }_0\ne 0,$${\beta }_0={\displaystyle \frac{1}{{\mathcal{R}}_{0,q}}}$ and ${}_{\mathcal{L}}\mathcal{H}_n^{(m,s)}(r_1,r_2,r_3),n=0,1,2,\dots ,$ are the mth-order three-variable Laguerre–Hermite polynomials.

Proof. 

By inserting the series forms of the new generalization of the three-variable Laguerre–Gould–Hopper polynomials into the generating function of the three-variable Laguerre–Gould–Hopper-based Appell polynomials, we obtain

$$\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(59)

By multiplying

$${\displaystyle \frac{1}{\mathcal{R}\left(t\right)}}=\sum _{k=0}^{\infty }{\beta }_k{\displaystyle \frac{t^k}{k!}},$$

(60)

on both sides, it follows that

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{k=0}^{\infty }{\beta }_k{\displaystyle \frac{t^k}{k!}}\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(61)

Applying Cauchy product in (61) gives

$${}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\beta }_k{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n-k}}^{(m,s)}(r_1,r_2,r_3).$$

(62)

This equality leads to a system of n equations with the unknowns ${\mathcal{R}}_n(r_1,r_2,r_3)$, where $n=0,1,2,\dots$

To solve this system using Cramer’s rule, we note that the denominator is the determinant of a lower triangular matrix, which has a determinant of ${\left({\beta }_0\right)}^{n+1}$. By taking the transpose of the numerator and replacing the $i^{th}$ row with the $(i+1)$th position for $i=1,2,\dots ,n-1$, we obtain the desired result. □

We will now demonstrate the multiplicative and derivative operators of ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$. The following theorem is presented:

Theorem 11. 

The generalization of mth-order Laguerre–Gould–Hopper-based Appell polynomials satisfies the multiplicative and derivative operators as follows:

$$\widehat{M}=r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}+r_{2}s{\widehat{D}}_{r_1}^{s-1}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1},$$

(63)

and

$$\widehat{P}={\widehat{D}}_{r_1},$$

(64)

respectively.

Proof. 

Utilizing the derivative with respect to t on both sides of Equation (54), we find

$$\begin{array}{c}\sum _{n=0}^{\infty }{}_{\mathcal{L}\mathcal{H}}{{\mathcal{R}}_{n+1}}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}}\mathcal{R}\left(t\right)e^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)+r_1\mathcal{R}\left(t\right)e^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)\hfill \\ \hfill +r_{2}s{t}^{s-1}\mathcal{R}\left(t\right)e^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)+m{\widehat{D}}_{r_3}^{-1}{t}^{m-1}\mathcal{R}\left(t\right)e^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m).\end{array}$$

(65)

Therefore, we have

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n+1}}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\left(r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}}+r_{2}s{t}^{s-1}+m{\widehat{D}}_{r_3}^{-1}{t}^{m-1}\right)\mathcal{R}\left(t\right)e^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)$$

By using Equation (54), we get

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n+1}}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left(r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}+r_{2}s{\widehat{D}}_{r_1}^{s-1}+m{D}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}\right){}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}$$

(66)

In view of (14) and (66), we get Assertion (63).

Again, differentiating Equation (54) with respect to t, it follows that

$${\widehat{D}}_{r_1}\left[\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right]=t\left[\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right],$$

which further can be written as

$${\widehat{D}}_{r_1}\left[\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}\right]=\left[\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^{n+1}}{n!}}\right].$$

(67)

Again in view of (15) and (67), we get Assertion (64). □

Theorem 12. 

The following differential equation for ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ holds true:

$$\left(r_1{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}+r_{2}s{\widehat{D}}_{r_1}^s+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^m-n\right){}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)=0.$$

(68)

Proof. 

Using (63) and (64) in (17), we get

$$\left(r_1{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}{\widehat{D}}_{r_1}+r_{2}s{\widehat{D}}_{r_1}^s+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^m\right){}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)=n{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3).$$

(69)

Upon the simplification, we get Assertion (68). □

The generalization of three-variable Hermite polynomials are defined by the following [16]:

$$e^{r_{1}t+r_2{t}^m+r_3{t}^s}=\sum _{n=0}^{\infty }{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(70)

Next, we prove the integral representations for the mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ in the following theorems.

Theorem 13. 

The following integral representation for the mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ holds true:

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)={\displaystyle \frac{1}{2\pi i}}{\int }_{-\infty }^{(0+)}{e}^u{u}^{-1}{}_{{\mathcal{H}}^{(m,s)}}{\mathcal{A}}_n(r_2,r_1{u}^{-1},r_3)du.$$

(71)

Proof. 

Using Equation (6) in the l.h.s. of Equation (33) and interchanging the sides, we have

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }{}_{\mathcal{L}}{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(72)

Using the following integral representation of LGHP ${}_{\mathcal{L}}{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)$ [8]:

$${}_{\mathcal{L}}{\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)={\displaystyle \frac{1}{2\pi i}}{\int }_{-\infty }^{(0+)}{e}^u{u}^{-1}{{\mathcal{H}}^{(m,s)}}_n(r_2,r_1{u}^{-1},r_3)du,$$

(73)

in the r.h.s. of Equation (72), we find

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{1}{2\pi i}}\mathcal{R}\left(t\right){\int }_{-\infty }^{(0+)}{e}^u{u}^{-1}\left(\sum _{n=0}^{\infty }{{\mathcal{H}}^{(m,s)}}_n(r_2,r_1{u}^{-1},r_3){\displaystyle \frac{t^n}{n!}}\right)du.$$

(74)

Now, making use of the generating function of generalized Hermite polynomials in the r.h.s. of the above equation, we have

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{1}{2\pi i}}{\int }_{-\infty }^{(0+)}{e}^u{u}^{-1}\left(\mathcal{R}\left(t\right)e^{r_{2}t+r_1{u}^{-1}{t}^m+r_3{t}^s}\right)du,$$

(75)

which in view of Equation (72) and comparing the coefficients of t on both sides of the above Equation (75), we get Assertion (71). □

Theorem 14. 

The following integral representations for the mth-order Laguerre-Hermite-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ holds true:

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)={\int }_0^{\infty }{e}^{-u}{}_{\mathcal{L}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,u{D}_{r_3}^{-1})du,$$

(76)

and

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)={\displaystyle \frac{1}{n!}}{\int }_0^{\infty }{e}^{-u}{u}^n{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m+1,s)}\left({\displaystyle \frac{r_1}{u}},r_2,r_3\right)du.$$

(77)

Proof. 

Using the integral representations [8]

$${}_{\mathcal{L}}\mathcal{H}^{(m,s)}(r_1,r_2,r_3)={\int }_0^{\infty }{e}^{-u}{}_{\mathcal{L}}{\mathcal{H}}^{(m,s)}(r_1,r_2,u{D}_{r_3}^{-1})du,$$

(78)

and

$${}_{\mathcal{L}}{\mathcal{H}}^{(m,s)}(r_1,r_2,r_3)={\displaystyle \frac{1}{n!}}{\int }_0^{\infty }{e}^{-u}{u}^n{}_{\mathcal{L}}{\mathcal{H}}^{(m+1,s)}\left({\displaystyle \frac{r_1}{u}},r_2,r_3\right)du,$$

(79)

respectively, on the l.h.s. of Equation (72), we find

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }\left({\int }_0^{\infty }{e}^{-u}{}_{\mathcal{L}}{\mathcal{H}}^{(m,s)}(r_1,r_2,u{D}_{r_3}^{-1})du\right){\displaystyle \frac{t^n}{n!}},$$

(80)

or, equivalently,

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}={\int }_0^{\infty }{e}^{-u}\left(\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }{}_{\mathcal{L}}{\mathcal{H}}^{(m,s)}(r_1,r_2,u{D}_{r_3}^{-1}){\displaystyle \frac{t^n}{n!}}\right)du,$$

(81)

and

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }\left({\displaystyle \frac{1}{n!}}{\int }_0^{\infty }{e}^{-u}{u}^n{}_{\mathcal{L}}{\mathcal{H}}^{(m+1,s)}\left({\displaystyle \frac{r_1}{u}},r_2,r_3\right)du\right){\displaystyle \frac{t^n}{n!}},$$

(82)

or, equivalently,

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{1}{n!}}{\int }_0^{\infty }{e}^{-u}{u}^n\left(\mathcal{R}\left(t\right)\sum _{n=0}^{\infty }{}_{\mathcal{L}}{\mathcal{H}}^{(m+1,s)}\left({\displaystyle \frac{r_1}{u}},r_2,r_3\right){\displaystyle \frac{t^n}{n!}}\right)du,$$

(83)

respectively.

Again, using Equation (72) in the l.h.s. of Equations (81) and (83), we find

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left({\int }_0^{\infty }{e}^{-u}{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,u{D}_{r_3}^{-1})du\right){\displaystyle \frac{t^n}{n!}},$$

(84)

and

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left({\displaystyle \frac{1}{n!}}{\int }_0^{\infty }{e}^{-u}{u}^n{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m+1,s)}\left({\displaystyle \frac{r_1}{u}},r_2,r_3\right)du\right){\displaystyle \frac{t^n}{n!}}.$$

(85)

respectively. Finally, equating the coefficients of like powers of t on both sides of Equations (84) and (85), we get Assertions (76) and (77). □

5. Generalized $\mathit{m}$th-Order Laguerre–Gould–Hopper-Based Appell Polynomials via Fractional Operators

In light of Euler’s integral identity, a powerful tool in fractional calculus and integral transforms (see [21,22]), we have

$$a^{-\nu }={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-at}{t}^{\nu -1}dt,min\{\mathrm{Re}\left(\nu \right),\mathrm{Re}\left(a\right)\}>0,$$

(86)

which allows us to represent inverse powers of linear operators in an integral form.

Applying this to specific differential operators leads to the operational formulas derived in [21]:

$${\left(\alpha -{\displaystyle \frac{d}{d{\mu }_1}}\right)}^{-\nu }f\left({\mu }_1\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}\left(e^{t{\displaystyle \frac{d}{d{\mu }_1}}}f\left({\mu }_1\right)\right)dt={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}f({\mu }_1+t)dt,$$

(87)

demonstrating how translation operators emerge naturally in this framework.

Similarly, when considering the second-order derivative operator, we obtain

$${\left(\alpha -{\displaystyle \frac{d^2}{d{{\mu }_1}^2}}\right)}^{-\nu }f\left({\mu }_1\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}\left(e^{t{\displaystyle \frac{d^2}{d{{\mu }_1}^2}}}f\left({\mu }_1\right)\right)dt,$$

(88)

which highlights the use of the heat kernel-type exponential operator in the analytical treatment of such inverse operators.

These integral representations not only facilitate the manipulation of fractional powers of differential operators but also form the foundation for developing operational approaches to special functions, including generalized polynomial families.

Theorem 15. 

For the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$, the following operational identity holds:

$${\left(\alpha -r_2{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}\right)}^{-\nu }{}_{{}_{\left[m\right]}\mathcal{L}}{\mathcal{R}}_n(r_1,r_3)={}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ).$$

(89)

Proof. 

Using Euler’s integral formula,

$$a^{-\nu }={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-at}{t}^{\nu -1}dt,min\{\mathrm{Re}\left(a\right),\mathrm{Re}\left(\nu \right)\}>0,$$

(90)

we substitute $a=\alpha -r_2{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}$. Then, we obtain

$$\begin{array}{cc}\hfill {\left(\alpha -r_2{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}\right)}^{-\nu }{}_{{}_{\left[m\right]}\mathcal{L}}{\mathcal{R}}_n(r_1,r_3)& ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}exp\left(t·r_2{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}\right){}_{{}_{\left[m\right]}\mathcal{L}}{\mathcal{R}}_n(r_1,r_3)dt\hfill \\ \hfill & ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,t{r}_2,r_3)dt.\hfill \end{array}$$

Define the mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ as

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ):={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,t{r}_2,r_3)dt,$$

(91)

which completes the proof. □

Theorem 16. 

The generating function for the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ is given by

$${\displaystyle \frac{\mathcal{R}\left(t\right)exp\left(r_{1}t\right)C_0(-r_3{t}^m)}{{\left(\alpha -r_2\frac{{\partial }^s}{{\partial }_{r_1^s}}\right)}^{\nu }}}=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}.$$

(92)

Proof. 

From the definition of the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3;\alpha )$, we have

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha s}{s}^{\nu -1}{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}ds.$$

(93)

Switching the order of summation and integration gives

$$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha s}{s}^{\nu -1}\left(\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,s{r}_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}\right)ds.$$

(94)

Using the known generating function of ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$:

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}=\mathcal{R}\left(t\right)exp\left(r_{1}t+r_{2}t{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}+{\widehat{D}}_{r_3}^{-1}{t}^m\right),$$

(95)

we get

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}& ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha s}{s}^{\nu -1}\mathcal{R}\left(t\right)exp\left(r_{1}t+r_{2}t{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}+{\widehat{D}}_{r_3}^{-1}{t}^m\right)ds\hfill \\ \hfill & =\mathcal{R}\left(t\right)exp\left(r_{1}t\right)C_0(-r_3{t}^m)·{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\left(\alpha -r_{2}t{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}\right)s}{s}^{\nu -1}ds.\hfill \end{array}$$

By Euler’s integral representation again, we conclude

$$\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}={\displaystyle \frac{\mathcal{R}\left(t\right)exp\left(r_{1}t\right)C_0(-r_3{t}^m)}{{\left(\alpha -r_{2}t\frac{{\partial }^s}{{\partial }_{r_1^s}}\right)}^{\nu }}}.$$

(96)

□

Remark 2. 

For $\alpha =1$, $\nu =1$, and $r_2={\widehat{D}}_{r_2}^{-1}$, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ reduce to the mth order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ (see Equation (54)).

The operational representation between the generalized mth -order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ and Appell polynomials $\mathcal{R}$ is obtained in the form of the following result:

Theorem 17. 

The following operational representation between the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ and Appell polynomials $\mathcal{R}$ holds true:

$${\left(\alpha -r_2{\displaystyle \frac{{\partial }^s}{{\partial }_{r_1^s}}}\right)}^{-\nu }exp\left(D_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{{\partial }_{r_1^m}}}\right){\mathcal{R}}_n\left(r_1\right)={}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ).$$

(97)

Proof. 

Through replacement of $r_2$ with $r_{2}t$ and multiplication by ${\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{e}^{-\alpha t}{t}^{\nu -1}$ and then integration with respect to t from $t=0$ to $t=\infty$ on Equation (54), we obtain

$${\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}exp\left({\widehat{D}}_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}+r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)\left\{{\mathcal{R}}_n\left(r_1\right)\right\}dt={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{e}^{-\alpha t}{t}^{\nu -1}{}_{{}_{\mathcal{L}}\mathcal{H}}{\mathcal{R}}_n(r_1,r_{2}t,r_3)dt.$$

(98)

Decoupling the exponential operator in the l.h.s. of the above-equation by using the Weyl identity [8],

$$e^{\widehat{A}+\widehat{B}}=e^{-k/2}{e}^{\widehat{A}}{e}^{\widehat{B}},k\in \mathbb{C},$$

(99)

we get

$$exp\left({\widehat{D}}_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}\right){\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}{\int }_0^{\infty }{t}^{\nu -1}{e}^{-\left(\alpha -r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)}dt{\mathcal{R}}_n\left(r_1\right)={}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$$

(100)

which, in view of relation (86), yields Assertion (97). □

We will now demonstrate the multiplicative and derivative operators of ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$. The following theorem is presented:

Theorem 18. 

The generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ satisfy the multiplicative and derivative operators as follows:

$$\widehat{M}=r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}+{\displaystyle \frac{\nu s{r}_2{\widehat{D}}_{r_1}^{s-1}}{\alpha -r_2{\widehat{D}}_{r_1}^s}},$$

(101)

and

$$\widehat{P}={\widehat{D}}_{r_1},$$

(102)

respectively.

Proof. 

Utilizing the derivative with respect to t on both sides of Equation (92), we find

$${\displaystyle \frac{\partial }{\partial t}}\left[{\displaystyle \frac{\mathcal{R}\left(t\right)exp\left(r_{1}t\right)C_0(-r_3{t}^m)}{{\left(\alpha -r_2{t}^s\right)}^{\nu }}}\right]={\displaystyle \frac{\partial }{\partial t}}\left[\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}\right],$$

(103)

which further gives

$$\left(r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}}+m{\widehat{D}}_{r_3}^{-1}{t}^{m-1}+{\displaystyle \frac{\nu s{r}_2{t}^{s-1}}{\alpha -r_2{t}^s}}\right)\left[{\displaystyle \frac{\mathcal{R}\left(t\right)exp\left(r_{1}t\right)C_0(-r_3{t}^m)}{{\left(\alpha -r_2{t}^s\right)}^{\nu }}}\right]=\sum _{n=0}^{\infty }n{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^{n-1}}{n!}}.$$

(104)

By using Equation (92), we get

$$\begin{array}{c}\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n+1,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}=\sum _{n=0}^{\infty }\left(r_1+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^{m-1}+{\displaystyle \frac{\nu s{r}_2{D}_{r_1}^{s-1}}{\alpha -r_2{\widehat{D}}_{r_1}^s}}\right)\hfill \\ \hfill \times {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}.\end{array}$$

(105)

In view of (14) and (105), we get Assertion (101).

Also, differentiating Equation (92) with respect to $r_1$, it follows that

$${\displaystyle \frac{\partial }{\partial r_1}}\left[{\displaystyle \frac{\mathcal{R}\left(t\right)exp\left(r_{1}t\right)C_0(-r_3{t}^m)}{{\left(\alpha -r_2{t}^s\right)}^{\nu }}}\right]=t\left[{\displaystyle \frac{\mathcal{R}\left(t\right)exp\left(r_{1}t\right)C_0(-r_3{t}^m)}{{\left(\alpha -r_2{t}^s\right)}^{\nu }}}\right],$$

which further can be written as

$${\displaystyle \frac{\partial }{\partial r_1}}\left[\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}\right]=\left[\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^{n+1}}{n!}}\right].$$

(106)

Again in view of (15) and (106), we get Aassertion (102). □

Theorem 19. 

The following differential equation for ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ holds true:

$$\left(r_1{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}{\widehat{D}}_{r_1}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^m+{\displaystyle \frac{\nu s{r}_2{\widehat{D}}_{r_1}^s}{\alpha -r_2{\widehat{D}}_{r_1}^s}}-n\right){}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )=0.$$

(107)

Proof. 

Using (101) and (102) in (17), we get

$$\left(r_1{\widehat{D}}_{r_1}+{\displaystyle \frac{{\mathcal{R}}^{\prime }\left({\widehat{D}}_{r_1}\right)}{\mathcal{R}\left({\widehat{D}}_{r_1}\right)}}{\widehat{D}}_{r_1}+m{\widehat{D}}_{r_3}^{-1}{\widehat{D}}_{r_1}^m+{\displaystyle \frac{\nu s{r}_2{\widehat{D}}_{r_1}^s}{\alpha -r_2{\widehat{D}}_{r_1}^s}}\right){}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )=n{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ).$$

(108)

Upon the simplification, we get Assertion (107). □

Next, we will demonstrate the determinant form for ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ using an approach similar to that presented in [18,19], taking into account Equation (54).

Theorem 20. 

The determinant representation of generalized mth-order three-variable Laguerre–Gould–Hopper–Appell polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{\mathrm{n},\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )$ of degree n is

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{0,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )={\displaystyle \frac{1}{{\beta }_0}},$$

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{n,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha )={\displaystyle \frac{{(-1)}^n}{{\left({\beta }_0\right)}^{n+1}}}$$

$$\times \left|\begin{array}{cccccc}1& {}_{{}_{\mathcal{L}}\mathcal{H}}_{1,\nu }^{(m,s)}(r_1,r_2,r_3;\alpha )& {}_{{}_{\mathcal{L}}\mathcal{H}}_{2,\nu }(r_1,r_2,r_3;\alpha )& \dots & {}_{{}_{\mathcal{L}}\mathcal{H}}_{n-1,\nu }^{(m,s)}(r_1,r_2,r_3;\alpha )& {}_{{}_{\mathcal{L}}\mathcal{H}}_{n,\nu }^{(m,s)}(r_1,r_2,r_3;\alpha )\\ {\beta }_0& {\beta }_1& {\beta }_2& \dots & {\beta }_{n-1}& {\beta }_n\\ 0& {\beta }_0& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\beta }_1& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-2}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\beta }_{n-1}\\ 0& 0& {\beta }_0& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\beta }_{n-3}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right){\beta }_{n-2}\\ ⋮& ⋮& ⋮& \dots & ⋮& \\ 0& 0& 0& \dots & {\beta }_0& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\beta }_1\end{array}\right|,$$

(109)

$${\beta }_n=-{\displaystyle \frac{1}{{\mathcal{R}}_0}}\left(\sum _{k=1}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\mathcal{R}}_k{\beta }_{n-k}\right),n=0,1,2,\dots ,$$

where ${\beta }_0\ne 0,$${\beta }_0={\displaystyle \frac{1}{{\mathcal{R}}_{0,q}}}$ and ${}_{{}_{\mathcal{L}}\mathcal{H}}_{n,\nu }^{(m,s)}(r_1,r_2,r_3;\alpha ),n=0,1,2,\dots ,$ are the generalized mth-order three-variable Laguerre–Gould–Hopper polynomials.

Proof. 

By substituting $r_3$ with $r_1$ and applying the operator

$${\left(\alpha -r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)}^{-\nu }exp\left(D_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}\right)$$

to both sides of Equations (8) and (97), on the right-hand side and left-hand side, respectively, we arrive at the following result:

$${}_{{}_{\mathcal{L}}\mathcal{H}}_{n,\nu }^{(m,s)}(r_1,r_2,r_3;\alpha )=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\beta }_{n-k}{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_{k,\nu }}^{(m,s)}(r_1,r_2,r_3;\alpha ){\displaystyle \frac{t^n}{n!}}.$$

(110)

This identity yields a system of n linear equations in terms of the unknowns ${\mathcal{R}}_n(r_1,r_2,r_3)$, where $n=0,1,2,\dots$.

To solve this system via Cramer’s rule, observe that the denominator corresponds to the determinant of a lower triangular matrix, whose value is ${\left({\beta }_0\right)}^{n+1}$. The solution is obtained by transposing the numerator matrix and substituting the ith row with the entry at position $(i+1)$th, for $i=1,2,\dots ,n-1$, resulting in the final expression. □

6. Examples

The Appell polynomial framework, driven by the function $\mathcal{R}\left(t\right)$, offers a flexible foundation for generating diverse polynomial families tailored to various scientific applications. Their analytical forms, numerical evaluations, and differential properties make them instrumental in modeling and computation across disciplines.

The Bernoulli, Euler, and Genocchi numbers are foundational in mathematics, with applications in number theory, combinatorics, algebraic geometry, and more. Bernoulli numbers appear in polynomials and the Euler–Maclaurin formula, while Euler numbers contribute to modular forms and elliptic curve theory. Genocchi numbers play a key role in combinatorial problems, graph theory, and automata theory. These numbers connect to hyperbolic secant functions and have implications in quantum field theory and signal processing. By treating them as members of the Appell family, new polynomials like the two-iterated Gould–Hopper–Appell polynomials are derived, offering rich research opportunities in their generating expressions and characteristics. As a result, different members of ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathcal{R}}_n}^{(m,s)}(r_1,r_2,r_3)$ appear as Laguerre–Gould–Hopper-based Bernoulli polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_n}^{(m,s)}(r_1,r_2,r_3)$, Laguerre–Gould–Hopper–Euler polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_n}^{(m,s)}(r_1,r_2,r_3)$, and Laguerre–Gould–Hopper–Genocchi polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{G}}_n}^{(m,s)}(r_1,r_2,r_3)$. The following expressions can be used to cast these polynomials:

$${\displaystyle \frac{t}{e^t-1}}{e}^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}},$$

(111)

$${\displaystyle \frac{2}{e^t+1}}{e}^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}},$$

(112)

and

$${\displaystyle \frac{2t}{e^t+1}}{e}^{r_{1}t+r_2{t}^s}{C}_0(-r_3{t}^m)=\sum _{n=0}^{\infty }{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{G}}_n}^{(m,s)}(r_1,r_2,r_3){\displaystyle \frac{t^n}{n!}}.$$

(113)

For instance, the mth-order Laguerre–Gould–Hopper-based Bernoulli polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_n}^{(m,s)}(r_1,r_2,r_3)$, Laguerre–Gould–Hopper–Euler polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_n}^{(m,s)}(r_1,r_2,r_3)$, and Laguerre–Gould–Hopper–Genocchi polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{G}}_n}^{(m,s)}(r_1,r_2,r_3)$ are defined by the following operational identities:

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_n}^{(m,s)}(r_1,r_2,r_3)=exp\left({\widehat{D}}_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}\right)\left\{{}_H{\mathcal{B}}_n^{\left(s\right)}(r_1,r_2)\right\}=exp\left(r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)\left\{{}_{{}_{\left[m\right]}\mathcal{L}}{\mathbb{B}}_n(r_1,r_3)\right\}$$

(114)

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_n}^{(m,s)}(r_1,r_2,r_3)=exp\left({\widehat{D}}_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}\right)\left\{{}_H{\mathcal{E}}_n^{\left(s\right)}(r_1,r_2)\right\}=exp\left(r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)\left\{{}_{{}_{\left[m\right]}\mathcal{L}}{\mathbb{E}}_n(r_1,r_3)\right\},$$

(115)

and

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{G}}_n}^{(m,s)}_{\mathrm{}}^{\mathrm{}}(r_1,r_2,r_3)=exp\left({\widehat{D}}_{r_3}^{-1}{\displaystyle \frac{{\partial }^m}{\partial r_1^m}}\right)\left\{{}_H{\mathcal{G}}_n^{\left(s\right)}(r_1,r_2)\right\}=exp\left(r_2{\displaystyle \frac{{\partial }^s}{\partial r_1^s}}\right)\left\{{}_{{}_{\left[m\right]}\mathcal{L}}{\mathbb{G}}_n(r_1,r_3)\right\}.$$

(116)

We investigate the beautiful zeros of the mth-order three-variable Laguerre–Gould–Hopper-based Bernoulli ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_n}^{(m,s)}(r_1,r_2,r_3)$, and Euler polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_n}^{(m,s)}(r_1,r_2,r_3)$, by using mathematica software.

A few of them are

$$\begin{array}{l}{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_0}^{(3,2)}(r_1,r_2,r_3)=1,\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_1}^{(3,2)}(r_1,r_2,r_3)=r_1-{\displaystyle \frac{1}{2}},\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_2}^{(3,2)}(r_1,r_2,r_3)=-r_1+r_1^2+{\displaystyle \frac{1}{2}}+2r_3-1,\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_3}^{(3,2)}(r_1,r_2,r_3)=-r_1+r_3^3+1+6r_3{r}_1+6r_2\\ +{\displaystyle \frac{3}{2}}{r}_1-{\displaystyle \frac{3}{2}}{r}_1^2-3r_3-{\displaystyle \frac{1}{4}},\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_4}^{(3,2)}(r_1,r_2,r_3)=-r_1+r_1^4+1+24r_1{r}_2+{\displaystyle \frac{3}{2}}-3r_1+3r_1^2\\ +6r_3+6r_3^2-2r_3{r}_1+12r_3{r}_1^2-12r_2+{\displaystyle \frac{8}{9}}\\ -4r_3-3+2r_1-2r_1^2-2r_1^3-{\displaystyle \frac{1}{5}}.\end{array}$$

Furthermore, a few of them are

$$\begin{array}{l}{}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_0}^{(5,2)}(r_1,r_2,r_3)=1,\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_1}^{(5,2)}(r_1,r_2,r_3)=-{\displaystyle \frac{1}{2}}+r_1,\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_2}^{(5,2)}(r_1,r_2,r_3)=r_1^2+2r_3-r_1,\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_3}^{(5,2)}(r_1,r_2,r_3)=-1+r_1^3-{\displaystyle \frac{3}{4}}-3r_3+{\displaystyle \frac{3}{2}}{r}_1+6r_3{r}_1\\ +6r_2-{\displaystyle \frac{3}{2}}{r}_1-{\displaystyle \frac{3}{2}}{r}_1^2,\\ {}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_4}^{(5,2)}(r_1,r_2,r_3)=r_1^4+6r_3-3r_1-6r_3{r}_1\\ -12r_2+24r_1{r}_2+6r_3^2-3r_1^2\\ -6r_3+6r_1+6r_1^2+2r_3{r}_2^2-2r_1-2r_1^3.\end{array}$$

Key properties of these polynomials can be further explored using the monomiality principle, which reveals structural insights. Their explicit forms provide clarity on term-wise behavior, while associated differential equations help establish deeper analytical connections. Additionally, studying determinant representations links these polynomials to matrix theory, offering new avenues for theoretical development and application.

Furthermore, in view of Expression (58), the polynomials ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_n}^{(m,s)}(r_1,r_2,r_3)$, ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_n}^{(m,s)}(r_1,r_2,r_3)$, and ${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{G}}_n}^{(m,s)}(r_1,r_2,r_3)$ satisfy the following determinant representations:

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{B}}_n}^{(m,s)}(r_1,r_2,r_3)={(-1)}^n\left|\begin{array}{cccccc}1& {}_{{}_{\mathcal{L}}\mathcal{H}}_1^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_2^{(m,s)}(r_1,r_2,r_3)& \dots & {}_{{}_{\mathcal{L}}\mathcal{H}}_{n-1}^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)\\ \\ 1& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{3}}& \dots & {\displaystyle \frac{1}{n}}& {\displaystyle \frac{1}{n+1}}\\ \\ 0& 1& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\displaystyle \frac{1}{2}}& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\displaystyle \frac{1}{n-1}}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\displaystyle \frac{1}{n}}\\ \\ 0& 0& 1& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{2}}\right){\displaystyle \frac{1}{n-2}}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right){\displaystyle \frac{1}{n-1}}\\ .& .& .& \dots & .& .\\ .& .& .& \dots & .& .\\ 0& 0& 0& \dots & 1& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\displaystyle \frac{1}{2}}\end{array}\right|,$$

(117)

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{E}}_n}^{(m,s)}(r_1,r_2,r_3)={(-1)}^n\left|\begin{array}{cccccc}1& {}_{{}_{\mathcal{L}}\mathcal{H}}_1^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_2^{(m,s)}(r_1,r_2,r_3)& \dots & {}_{{}_{\mathcal{L}}\mathcal{H}}_{n-1}^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)\\ \\ 1& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}{\left(1\right)}_2& \dots & {\displaystyle \frac{1}{2}}{\left(1\right)}_{n-1}& {\displaystyle \frac{1}{2}}{\left(1\right)}_n\\ \\ 0& 1& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\displaystyle \frac{1}{2}}& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\displaystyle \frac{1}{2}}{\left(1\right)}_{n-2}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\displaystyle \frac{1}{2}}{\left(1\right)}_{n-1}\\ \\ 0& 0& 1& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{2}}\right){\displaystyle \frac{1}{2}}{\left(1\right)}_{n-3}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right){\displaystyle \frac{1}{2}}{\left(1\right)}_{n-2}\\ .& .& .& \dots & .& .\\ .& .& .& \dots & .& .\\ 0& 0& 0& \dots & 1& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\displaystyle \frac{1}{2}}\end{array}\right|,$$

(118)

and

$${}_{{}_{\mathcal{L}}\mathcal{H}}{{\mathbb{G}}_n}^{(m,s)}(r_1,r_2,r_3)={(-1)}^n\left|\begin{array}{cccccc}1& {}_{{}_{\mathcal{L}}\mathcal{H}}_1^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_2^{(m,s)}(r_1,r_2,r_3)& \dots & {}_{{}_{\mathcal{L}}\mathcal{H}}_{n-1}^{(m,s)}(r_1,r_2,r_3)& {}_{{}_{\mathcal{L}}\mathcal{H}}_n^{(m,s)}(r_1,r_2,r_3)\\ \\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{{\left(1\right)}_2}{4}}& {\displaystyle \frac{{\left(1\right)}_3}{6}}& \dots & {\displaystyle \frac{{\left(1\right)}_n}{2n}}& {\displaystyle \frac{{\left(1\right)}_{n+1}}{2(n+1)}}\\ \\ 0& {\displaystyle \frac{1}{2}}& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{1}}\right){\displaystyle \frac{{\left(1\right)}_2}{4}}& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{1}}\right){\displaystyle \frac{{\left(1\right)}_{n-1}}{2(n-1)}}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{1}}\right){\displaystyle \frac{{\left(1\right)}_n}{2n}}\\ \\ 0& 0& {\displaystyle \frac{1}{2}}& \dots & \left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{2}}\right){\displaystyle \frac{{\left(1\right)}_{n-2}}{2(n-2)}}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right){\displaystyle \frac{{\left(1\right)}_{n-1}}{2(n-1)}}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & {\displaystyle \frac{1}{2}}& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right){\displaystyle \frac{{\left(1\right)}_2}{4}}\end{array}\right|.$$

(119)

7. Concluding Remarks

In this work, we introduced a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and thoroughly investigated their core properties. Utilizing the concept of quasi-monomiality, we derived fundamental results such as recurrence relations, operational formulas, and the associated differential equations. We also provided both series and determinant representations to facilitate further analysis. Several important subfamilies, including the generalized mth-order Laguerre–Hermite Appell polynomials, were studied in detail. Additionally, we extended the framework to define the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials via fractional operators and introduced new families like the Laguerre–Gould–Hopper–Bernoulli, Laguerre–Gould–Hopper–Euler, and Laguerre–Gould–Hopper–Genocchi polynomials.

For future research, exploring orthogonality relations and weight functions associated with these polynomial families could provide deeper insight. Developing q-analogues and multivariate extensions also presents promising directions, potentially linking these polynomials to broader areas such as combinatorics and quantum calculus. Applications to differential equations and mathematical physics remain an exciting avenue for further study.