Certain Properties and Characterizations of Multivariable Hermite-Based Appell Polynomials via Factorization Method

Abstract

This paper introduces a new type of polynomials generated through the convolution of generalized multivariable Hermite polynomials and Appell polynomials. The paper explores several properties of these polynomials, including recurrence relations, explicit formulas using shift operators, and differential equations. Further, integrodifferential and partial differential equations for these polynomials are also derived. Additionally, the study showcases the practical applications of these findings by applying them to well-known polynomials, such as generalized multivariable Hermite-based Bernoulli and Euler polynomials. Thus, this research contributes to advancing the understanding and utilization of these hybrid polynomials in various mathematical contexts.

1. Introduction and Preliminaries

To illustrate the core concept of the factorization method, we briefly touch upon the widely discussed comparison between Maxwell’s and Dirac’s equations. Both systems, referring to Maxwell’s equations and the specific linear systems being compared, share common features, such as the involvement of first-order partial derivatives and the property of Lorentz invariance. However, it is essential to recognize a notable distinction arising from the linearity of Maxwell’s equations. This linearity can give rise to challenges associated with infinite self-energies. In simpler terms, while both systems exhibit similar mathematical characteristics and Lorentz invariance, the specific issue of infinite self-energies arises uniquely in the context of Maxwell’s equations due to their linear nature.

The factorization method [1] is a valuable approach employed by physicists to address eigenvalue problems. This practical technique involves the solution of two first-order differential equations, which, when combined, result in a second-order differential equation of equal importance. In addition to that, the factorization method includes the calculation of transition probabilities while taking into account the production process. Moreover, it provides a comprehensive framework for effectively addressing perturbation issues. In simpler terms, this technique utilizes the solution of two specific types of differential equations to obtain another equation that is equally important. It goes beyond simple computation by incorporating transition probabilities, which consider how a system changes over time. This method also offers a flexible approach to handle perturbation concerns, which are disruptions or disturbances that may affect the stability or accuracy of the system being studied.

We consider a sequence of polynomials denoted as ${{\mathcal{P}}_n\left(j_1\right)}_{n=0}^{\infty }$, where n is a non-negative integer representing the degree of the polynomials. Within this context, we have two sequences of differential operators, ${\mathsf{\Psi }}_n^{+}$ and ${\mathsf{\Psi }}_n^{-}$, which act upon this polynomial sequence. These operators possess specific properties:

$${\mathsf{\Psi }}_n^{+}\left({\mathcal{P}}_n\left(j_1\right)\right)={\mathcal{P}}_{n+1}\left(j_1\right)$$

(1)

and

$${\mathsf{\Psi }}_n^{-}\left({\mathcal{P}}_n\left(j_1\right)\right)={\mathcal{P}}_{n-1}\left(j_1\right).$$

(2)

An important property known as the differential equation is expressed as follows:

$${\mathcal{P}}_n\left(j_1\right)=\left({\mathsf{\Psi }}_{n+1}^{-}{\mathsf{\Psi }}_n^{+}\right)\left\{{\mathcal{P}}_n\left(j_1\right)\right\}.$$

(3)

This equation is derived by utilizing the operators ${\mathsf{\Psi }}_n^{-}$ and ${\mathsf{\Psi }}_n^{+}$. The expression (3) is used as a starting point to construct differential equations using the factorization method. The major goal of using this method is to identify two distinct operators: the multiplicative operator ${\mathsf{\Psi }}_n^{+}$ and the derivative operator ${\mathsf{\Psi }}_n^{-}$. The selection of these operators ensures that they meet the conditions in Equation (3). The original Equation (3) is converted into a series of differential equations using the derived operators ${\mathsf{\Psi }}_n^{-}$ and ${\mathsf{\Psi }}_n^{+}$ by using the factorization approach. Finding appropriate formulations for these operators that fulfill the provided equation is the goal. By using these operators to reframe the issue, we may obtain new perspectives and simplify our understanding of how to analyze and solve the equation. These differential equations may be built methodically using the factorization methodology, and the correct operators that correspond to Equation (3) can be identified. It enables a more methodical and narrowed approach to comprehending and working with the provided equation.

The operational rule for the 2-V Hermite Kampé de Fériet polynomials (2VHKdFP) ${\mathcal{Y}}_n^{\left[2\right]}(j_1,j_2)$ is described in [2]

$${\mathcal{Y}}_n^{\left[2\right]}(j_1,j_2)=exp\left(j_2\frac{{\partial }^2}{\partial {j_1}^2}\right){j_1}^n.$$

(4)

The operational rule acts as a guideline or process for carrying out particular operations or computations linked to these polynomials in their study of them. It explains how to alter or assess 2VHKdFP by using certain mathematical operations or transformations.

These polynomials are specified by the generating expression:

$$\sum _{n=0}^{\infty }{\mathcal{Y}}_n^{\left[2\right]}(j_1,j_2)\frac{{\xi }^n}{n!}=exp(j_1\xi +j_2{\xi }^2)$$

(5)

and represented by series:

$${\mathcal{Y}}_n^{\left[2\right]}(j_1,j_2)=n!\sum _{k=0}^{\left[\frac{n}{2}\right]}\frac{j_2^k{j}_1^{n-2k}}{k!(n-2k)!}.$$

(6)

Further, the polynomials ${\mathcal{Y}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)$, known as multivariable Hermite polynomials (MHP) [3], are given by relation:

$$exp(j_1\xi +j_2{\xi }^2+\cdots +j_m{\xi }^m)=\sum _{n=0}^{\infty }{\mathcal{Y}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!},$$

(7)

with the operational rule as

$$exp\left(j_2\frac{{\partial }^2}{\partial {j_1}^2}+j_3\frac{{\partial }^3}{\partial {j_1}^3}+\cdots +j_m\frac{{\partial }^m}{\partial {j_1}^m}\right)j_1^n={\mathcal{Y}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m),$$

(8)

and series representation as

$${\mathcal{Y}}_n^{\left[m\right]}(j_1,j_2,\cdots j_m)=n!\sum _{r=0}^{[n/m]}\frac{j_m^r{\mathcal{Y}}_{n-mr}^{\left[m\right]}(j_1,j_2,\cdots ,j_{m-1})}{r!(n-mr)!}.$$

(9)

In comparison with other polynomial sequences, Appell polynomial sequences [4] are very important and are used in many different areas of mathematics, including engineering sciences, theoretical physics, engineering mathematics, medical sciences, and approximation theory. Additionally, when combined, Appell polynomials satisfy each of the axioms of an abelian group. They also have several uses in number theory, algebraic geometry, combinatorics, and other areas of mathematics. There are many uses for these polynomials; for instance, solutions to partial differential equations are expressed in terms of power series using Appell polynomials. Particular Appell polynomials with significant uses in mathematical physics are the Hermite, Laguerre, and Legendre polynomials. Additionally, the study of algebraic surfaces and curves makes use of these polynomials. They offer a method for creating unique algebraic curve divisors, which is helpful for examining their characteristics. Additionally, in combinatorics, Appell polynomials are used to count numerous objects, including set partitions and graphs. In the investigation of symmetric functions and their characteristics, they are also employed. As a result, number theory uses these polynomials to explore modular forms and modular curves. As linear combinations of modular forms, they offer a means of expressing modular shapes.

A number of polynomial sequences were first presented by Appell in the 19th century. These polynomial sequences are connected to one another, and their connection may be mathematically described. The Appell polynomials are characterized by this connection, which is fundamental to many areas of mathematics and holds the differential expression [4]

$$\frac{d}{d{j}_1}{\mathfrak{A}}_k\left(j_1\right)=k{\mathfrak{A}}_{k-1}\left(j_1\right),k\in {\mathbb{N}}_0$$

(10)

and generating relation

$$\mathfrak{A}\left(\xi \right)e^{j_1\xi }=\sum _{k=0}^{\infty }{\mathfrak{A}}_k\left(j_1\right)\frac{{\xi }^k}{k!},$$

(11)

where $\mathfrak{A}\left(\xi \right)$ is a mathematical function defined on the real line. As the input values become closer to a certain value, this function converges, which implies that it approaches a finite limit. Additionally, the mathematical formulation for this function’s Taylor expansion is

$$\mathfrak{A}\left(\xi \right)=\sum _{k=0}^{\infty }{\mathfrak{A}}_k\frac{{\xi }^k}{k!},{\mathfrak{A}}_0\ne 0.$$

(12)

It is clear that distinct characteristics of hybrid special polynomials are developed by the integration of numerous concepts, including the monomiality principle, operational rules, and other qualities. The idea of monomiality has found widespread applications in many mathematical fields, including mathematical physics, quantum mechanics, and classical optics. It was first proposed by Steffenson as poweriods in 1941 and later developed by Dattoli [5,6]. As demonstrated by works such as [7,8,9,10,11,12], these operational approaches are useful and successful research tools. The study and use of hybrid special polynomials in numerous areas of mathematics have greatly benefited from the adoption of these ideas.

The factorization approach, pioneered by He and Ricci [13], has been widely used to obtain differential equations for Appell polynomials and their multivariable extensions, as well as for Bernoulli and Euler polynomials, as described in [14,15]. Furthermore, the method has been extended to derive expressions such as integrodifferential and partial differential for hybrid forms, 2D extended, and mixed-type Appell family polynomials, as seen in [16,17]. In a study by Ozarslan [18], the Appell polynomials were employed to generate a series of finite-order differential equations through the expansion of the factorization technique using k-times shift operators. These findings have been instrumental in establishing recurrence relations, shift operators, and families of differential equations for multivariable Hermite Appell polynomials, as indicated by Equation (13) in the current article. The factorization approach is a valuable tool for deriving equations for different types of polynomials across various fields of mathematics.

The development of various operational techniques, recurrence relations, shift operators, and families of differential equations for different types of polynomials in several areas of mathematics has motivated the construction of multivariable Hermite–Appell polynomials (M${}_D$HAP). These polynomials are denoted by ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ and are generated by the specific expression

$$\mathfrak{A}\left(\xi \right)exp(j_1\xi +j_2{\xi }^2+j_3{\xi }^3+\cdots +j_m{\xi }^m)=\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\frac{{\xi }^n}{n!},$$

(13)

where $\mathfrak{A}\left(\xi \right)$ is given by (12). The generating expression used to construct these polynomials involves the application of a suitable linear operator on the product of m Hermite polynomials. The rest of the article is listed as

This article explores the properties and characteristics of multivariable Hermite-based Appell polynomials. It focuses on constructing families of differential equations for these polynomials using the factorization approach. For these polynomials, Section 2 discusses the generating relation, recurrence relation, and shift operators. The development of several families of differential equations for these polynomials is covered in Section 3. The article also discusses these polynomials’ specific instances, such as the Bernoulli, Euler, and Genocchi polynomials; solutions for these polynomials are found in Section 4. In the concluding part, we summarize the findings of the article.

2. Recurrence Relations and Shift Operators

Deriving the shift operators and recurrence relations for M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,$$j_m)$ is the major objective of this section. The relations between different instances of M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ and different values of the indices n, $j_1$, $j_2$, $j_3$, and so on are created throughout the derivation process. These recurrence relations may be used to represent the polynomials in terms of one another, which can be useful for accelerating computations or identifying recurrent patterns. By establishing these recurrence relations and shift operators, it is able to better understand the properties and behaviors of M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$. In the appropriate field of study, this knowledge may be helpful for a variety of computations, analyses, or applications involving these polynomials. The recurrence relation for the function ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ is constructed using the following result:

Theorem 1.

M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ fulfill the following recurrence relation:

$$\begin{array}{c}{}_{\mathcal{Y}}{\mathfrak{A}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\equiv (j_1+i_0){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)i_k{}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+\hfill \\ 2n{j}_2{}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+3n(n-1)j_3{}_{\mathcal{Y}}{\mathfrak{A}}_{n-2}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+\cdots \cdots \cdots +n(n-1)(n-2)\\ \hfill \times \cdots (n-m+1)j_m{}_{\mathcal{Y}}{\mathfrak{A}}_{n-m}^{\left[m\right]}(j_1,j_2,\cdots ,j_m),\end{array}$$

(14)

where the coefficients ${\left\{i_k\right\}}_{k\in {\mathbb{N}}_0}$ are represented by

$$\frac{{\mathfrak{A}}^{\prime }\left(\xi \right)}{\mathfrak{A}\left(\xi \right)}=\sum _{k=0}^{\infty }{i}_k\frac{{\xi }^k}{k!}.$$

(15)

Proof. 

Taking the derivatives of Equation (13) w.r.t. $\xi$, we find

$$\begin{array}{c}\hfill \frac{\partial }{\partial \xi }\left\{\mathfrak{A}\left(\xi \right)exp(j_1\xi +j_2{\xi }^2+\cdots +j_m{\xi }^m)\right\}=\left\{\frac{{\mathfrak{A}}^{\prime }\left(\xi \right)}{\mathfrak{A}\left(\xi \right)}+j_1+2j_2\xi +3j_3{\xi }^2+\cdots +m{j}_m{\xi }^{m-1}\right\}\\ \hfill \times \left\{\mathfrak{A}\left(\xi \right)exp(j_1\xi +j_2{\xi }^2+\cdots +j_m{\xi }^m)\right\}.\end{array}$$

(16)

By substituting the right-hand side (RHS) of expression (13) into the previous Equation (16), we obtain

$$\begin{array}{c}\hfill \frac{\partial }{\partial \xi }\left\{\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}\right\}=\left\{\frac{{\mathfrak{A}}^{\prime }\left(\xi \right)}{\mathfrak{A}\left(\xi \right)}+j_1+2j_2\xi +3j_3{\xi }^2+\cdots +m{j}_m{\xi }^{m-1}\right\}\\ \hfill \times \left\{\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}\right\}.\end{array}$$

(17)

Thus, in view of Equation (15), we can express the previous equation in the following form:

$$\begin{array}{c}\sum _{n=1}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^{n-1}}{(n-1)!}=\left(j_1+\sum _{k=0}^{\infty }{i}_k\frac{{\xi }^k}{k!}\right)\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}+2j_2\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1\hfill \\ \hfill ,j_2,\cdots ,j_m)\frac{{\xi }^{n+1}}{n!}+3j_3\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^{n+2}}{n!}+\cdots +m{j}_m\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^{n+m}}{n!}.\end{array}$$

(18)

A further simplification of the equation yields

$$\begin{array}{c}\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^{n-1}}{(n-1)!}=\left(j_1+i_0\right)\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}+\sum _{k=1}^{\infty }{i}_k\frac{{\xi }^k}{k!}\times \sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,\hfill \\ j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}+2j_2\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^{n+1}}{n!}+3j_3\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^{n+2}}{n!}+\cdots +\\ \hfill m{j}_m\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^{n+m}}{n!}.\end{array}$$

After replacing n with $n+1$ in the left-hand side (LHS) of the previous equation, and replacing n with $n-k$, $n-1$, $n-2$, and so on up to $n-m$ in the right-hand-side (RHS) terms, we obtain

$$\begin{array}{c}\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}=\left(j_1+i_0\right)\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}+\sum _{n=0}^{\infty }\sum _{k=1}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right){}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\hfill \\ \cdots ,j_m)\frac{{\xi }^n}{n!}+2j_2\sum _{n=0}^{\infty }n{}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}+3j_3\sum _{n=0}^{\infty }n(n-1){}_{\mathcal{Y}}{\mathfrak{A}}_{n-2}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}+\\ \hfill \cdots +m{j}_{m}n(n-1)(n-2)\cdots (n-m+1)\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{A}}_{n-m}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\frac{{\xi }^n}{n!}.\end{array}$$

(19)

By comparing the coefficients of corresponding exponents of $\frac{{\xi }^n}{n!}$ on both sides of Equation (19), we establish the validity of assertion (14).  □

In the following analysis, we present the construction of shift operators for M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)$ by establishing the subsequent result:

Theorem 2.

M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)$ satisfy the listed shift operators:

$${}_{j_1}{\pounds }_n^{-}:=\frac{1}{n}{D}_{j_1},$$

(20)

$${}_{j_2}{\pounds }_n^{-}:=\frac{1}{n}{D}_{j_1}^{-1}{D}_{j_2},$$

(21)

$${}_{j_3}{\pounds }_n^{-}:=\frac{1}{n}{D}_{j_1}^{-2}{D}_{j_3},$$

(22)

⋮

$${}_{j_m}{\pounds }_n^{-}:=\frac{1}{n}{D}_{j_1}^{-(m-1)}{D}_{j_m},$$

(23)

$${}_{j_1}{\pounds }_n^{+}:=(j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^k+2j_2{D}_{j_1}+3j_3{D}_{j_1}^2+\cdots +m{j}_m{D}_{j_1}^{m-1}$$

(24)

$${}_{j_2}{\pounds }_n^{+}:=(j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-(k-1)}{D}_{j_2}^{k-1}+2j_2{D}_{j_1}^{-1}{D}_{j_2}+3j_3{D}_{j_1}^{-2}{D}_{j_2}^2+\cdots +m{j}_m{D}_{j_1}^{-(m-1)}{D}_{j_2}^{m-1},$$

(25)

$${}_{j_3}{\pounds }_n^{+}:=(j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-2(k-1)}{D}_{j_3}^{k-1}+2j_2{D}_{j_1}^{-2}{D}_{j_3}+3j_3{D}_{j_1}^{-4}{D}_{j_3}^2+\cdots +m{j}_m{D}_{j_1}^{-2(m-1)}{D}_{j_3}^{m-1},$$

(26)

⋮

$$\begin{array}{cc}\hfill {}_{j_m}{\pounds }_n^{+}:=(j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-{(k-1)}^2}{D}_{j_m}^{k-1}+2j_2{D}_{j_1}^{-(m-1)}{D}_{j_m}+& \\ \hfill 3j_3{D}_{j_1}^{-2(m-1)}{D}_{j_m}^2+\cdots +m{j}_m{D}_{j_1}^{-{(m-1)}^2}{D}_{j_m}^{m-1},\end{array}$$

(27)

where

$$D_{j_1}:=\frac{\partial }{\partial j_1},D_{j_2}:=\frac{\partial }{\partial j_2},D_{j_3}:=\frac{\partial }{\partial j_3}\mathrm{and}{D}_{j_1}^{-1}:={\int }_0^{j_1}f\left(\eta \right)d\eta .$$

Proof. 

By differentiating Equation (13) with respect to $j_1$ and subsequently equating the coefficients of similar exponents of $\xi$ on both sides of the resulting equation, we obtain the following expression:

$$\frac{\partial }{\partial j_1}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=n{}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m).$$

(28)

As a consequence of the aforementioned steps, we arrive at the following expression:

$${}_{j_1}{\pounds }_n^{-}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=\frac{1}{n}{D}_{j_1}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}={}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m),$$

(29)

which establishes the validity of assertion (20).

By performing the differentiation of Equation (13) with respect to $j_2$ and equating the coefficients of the corresponding powers of $\xi$ on both sides, we obtain the following expression:

$$\frac{\partial }{\partial j_2}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=n(n-1){}_{\mathcal{Y}}{\mathfrak{A}}_{n-2}^{\left[m\right]}(j_1,j_2,\cdots ,j_m).$$

(30)

The previous expression can alternatively be expressed as

$$\frac{\partial }{\partial j_2}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=n\frac{\partial }{\partial j_1}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\},$$

(31)

thus eventually giving

$${}_{j_2}{\pounds }_n^{-}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=\frac{1}{n}{D}_{j_1}^{-1}{D}_{j_2}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}={}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m).$$

(32)

Therefore, the validity of assertion (21) is established.

After differentiating Equation (13) with respect to $j_3$ and equating the coefficients of the identical powers of $\xi$ on both sides, we obtain the following derived expression:

$$\frac{\partial }{\partial j_3}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=n(n-1)(n-2){}_{\mathcal{Y}}{\mathfrak{A}}_{n-3}^{\left[m\right]}(j_1,j_2,\cdots ,j_m).$$

(33)

The previous Equation (33) can also be expressed as

$$\frac{\partial }{\partial j_3}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=n\frac{{\partial }^2}{\partial j_1^2}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\},$$

(34)

thus eventually giving

$${}_{j_3}{\pounds }_n^{-}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=\frac{1}{n}{D}_{j_1}^{-2}{D}_{j_3}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}={}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m),$$

(35)

hence, yielding assertion (22).

Finally, upon differentiating Equation (13) with respect to $j_m$ and equating the coefficients of the same powers of $\xi$ on both sides of the resulting equation, we arrive at the following expression:

$$\frac{\partial }{\partial j_m}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=n(n-1)(n-2)(n-m+1){}_{\mathcal{Y}}{\mathfrak{A}}_{n-m}^{\left[m\right]}(j_1,j_2,\cdots ,j_m),$$

(36)

further presented as

$$\frac{\partial }{\partial j_m}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=n\frac{{\partial }^{m-1}}{\partial j_1^{m-1}}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}$$

(37)

and finally giving

$${}_{j_m}{\pounds }_n^{-}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}=\frac{1}{n}{D}_{j_1}^{-(m-1)}{D}_{j_m}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}={}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m).$$

(38)

Therefore, the validity of assertion (23) is established.

To establish the equation for the raising operator Equation (24), we employ the following expression:

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=({}_{j_1}{\pounds }_{n-k+1}^{-}{}_{j_1}{\pounds }_{n-k+2}^{-}\cdots {}_{j_1}{\pounds }_{n-1}^{-}{}_{j_1}{\pounds }_n^{-})\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(39)

Thus, in view of expression (29), expression (39) in a simplified form can be presented as

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=\frac{(n-k)!}{n!}{D}_{j_1}^k\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(40)

By substituting Equation (40) into the recurrence relation Equation (14), we deduce that

$$\begin{array}{cc}\hfill {}_{\mathcal{Y}}{\mathfrak{A}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=\left((j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^k+2j_2{D}_{j_1}+3j_3{D}_{j_1}^2+\cdots +m{j}_m{D}_{j_1}^{m-1}\right)& \\ \hfill \times {}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m).& \end{array}$$

(41)

Thus, Equation (24) of the raising operator ${}_{j_1}{\pounds }_n^{+}$ is proved.

In order to demonstrate the raising operator Equation (25), we examine the following relation:

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=({}_{j_2}{\pounds }_{n-k+1}^{-}{}_{j_2}{\pounds }_{n-k+2}^{-}\cdots {}_{j_2}{\pounds }_{n-1}^{-}{}_{j_2}{\pounds }_n^{-})\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(42)

By taking into account Equation (32), we can expand the above expression as follows:

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=\frac{(n-k)!}{k!}{D}_{j_1}^{-(k-1)}{D}_{j_2}^{(k-1)}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(43)

By substituting Equation (43) into the recurrence relation Equation (14), we can deduce that

$$\begin{array}{cc}\hfill {}_{\mathcal{Y}}{\mathfrak{A}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=((j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-(k-1)}{D}_{j_2}^{k-1}+2j_2{D}_{j_1}^{-1}{D}_{j_2}+3j_3{D}_{j_1}^{-2}{D}_{j_2}^2+& \\ \hfill \cdots +m{j}_m{D}_{j_1}^{-(m-1)}{D}_{j_2}^{m-1}).\end{array}$$

(44)

Therefore, we have successfully established the validity of assertion (25) for the raising operator ${}_{j_2}{\pounds }_n^{+}$.

To demonstrate the raising operator $j_3{\pounds }^{+}n$, we consider the following expression:

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=({}_{j_3}{\pounds }_{n-k+1}^{-}{}_{j_3}{\pounds }_{n-k+2}^{-}\cdots {}_{j_3}{\pounds }_{n-1}^{-}{}_{j_3}{\pounds }_n^{-})\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(45)

By taking into account Equation (35), we can expand the above expression as follows:

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=\frac{(n-k)!}{k!}{D}_{j_1}^{-2(k-1)}{D}_{j_3}^{(k-1)}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(46)

By substituting Equation (46) into the recurrence relation Equation (14), we find that

$$\begin{array}{cc}\hfill {}_{\mathcal{Y}}{\mathfrak{A}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=((j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-2(k-1)}{D}_{j_3}^{k-1}+2j_2{D}_{j_1}^{-2}{D}_{j_3}+3j_3{D}_{j_1}^{-4}{D}_{j_3}^2+& \\ \hfill \cdots +m{j}_m{D}_{j_1}^{-2(m-1)}{D}_{j_3}^{m-1}).\end{array}$$

(47)

Therefore, we have successfully established the validity of assertion (26) for the raising operator ${}_{j_3}{\pounds }_n^{+}$.

Finally, To establish the raising operator ${}_{j_m}{\pounds }_n^{+}$, we analyze the following expression:

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=({}_{j_m}{\pounds }_{n-k+1}^{-}{}_{j_m}{\pounds }_{n-k+2}^{-}\cdots {}_{j_m}{\pounds }_{n-1}^{-}{}_{j_m}{\pounds }_n^{-})\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(48)

By taking into account Equation (38), we can expand the above expression as follows:

$${}_{\mathcal{Y}}{\mathfrak{A}}_{n-k}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=\frac{(n-k)!}{k!}{D}_{j_1}^{-{(k-1)}^2}{D}_{j_m}^{(k-1)}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\right\}.$$

(49)

By substituting Equation (49) into the recurrence relation Equation (14), we deduce that

$$\begin{array}{cc}\hfill {}_{\mathcal{Y}}{\mathfrak{A}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=((j_1+i_0)+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-(k-1)}{D}_{j_m}^{k-1}+2j_2{D}_{j_1}^{-(m-1)}{D}_{j_m}+& \\ \hfill 3j_3{D}_{j_1}^{-2(m-1)}{D}_{j_m}^2+\cdots +m{j}_m{D}_{j_1}^{-{(m-1)}^2}{D}_{j_m}^{m-1}).\end{array}$$

(50)

Thus, expression (27) of the raising operator ${}_{j_m}{\pounds }_n^{+}$ is proved.  □

Further, it may be noted down that these raising and lowering operators satisfy the commutative relation:

$$\begin{array}{c}\left[\frac{\partial }{\partial \xi },\frac{\partial }{\partial j_1}\right]{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=\frac{\partial }{\partial \xi }\left(\frac{\partial }{\partial j_1}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\right\}\right)\hfill \\ -\frac{\partial }{\partial j_1}\left(\frac{\partial }{\partial \xi }\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\right\}\right)\\ =\frac{\partial }{\partial \xi }\left\{n{}_{\mathcal{Y}}{\mathfrak{A}}_{n-1}^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\right\}-\frac{\partial }{\partial j_1}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_{n+1}^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\right\}\\ =(n+1){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)-n{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\\ ={}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\end{array}$$

(51)

That is,

$$\left[\frac{\partial }{\partial \xi },\frac{\partial }{\partial j_1}\right]=1.$$

In the subsequent section, we delve into the examination of the families of differential equations that are fulfilled by the multivariable Hermite-based Appell polynomials. The section encompasses a comprehensive exploration of different categories of differential equations, encompassing differential, integrodifferential, and partial differential equations. These equations are derived through the application of the factorization method.

3. Families of Differential Equations

In this section, we provide comprehensive explanations of each type of equation, elucidating its structure and its relationship with the multivariable Hermite-based Appell polynomials. For the M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)$, we establish the differential, integrodifferential, and partial differential equations. Additionally, we construct the differential equation for MDHAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)$ by presenting the following conclusion:

Theorem 3.

M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ satisfy the following differential equation:

$$\left((j_1+i_0)D_{j_1}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{k+1}+2j_2{D}_{j_1}^2+3j_2{D}_{j_1}^3+\cdots +j_m{D}_{j_1}^m-n\right){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.$$

(52)

Proof. 

Expressions (20) and (24) of the shift operators are utilized in the factorization relation as follows:

$${}_{j_1}{\pounds }_{n+1}^{-}{}_{j_1}{\pounds }_n^{+}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\right\}={}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m).$$

(53)

Simplifying the mathematical expression, assertion (52) is proved.  □

Theorem 4.

M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ satisfy the following integrodifferential equations:

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_2}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-(k-1)}{D}_{j_2}^k+2j_2{D}_{j_1}^{-1}{D}_{j_2}^2+3j_3{D}_{j_1}^{-2}{D}_{j_2}^3+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-(m-1)}{D}_{j_2}^m-(n+1)D_{j_1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(54)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_3}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-(k-1)}{D}_{j_2}^{k-1}{D}_{j_3}+2j_2{D}_{j_1}^{-1}{D}_{j_2}{D}_{j_3}+3j_3{D}_{j_1}^{-2}{D}_{j_2}^2{D}_{j_3}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-(m-1)}{D}_{j_2}^{m-1}{D}_{j_3}-(n+1)D_{j_1}^2){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(55)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_m}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-(k-1)}{D}_{j_2}^{k-1}{D}_{j_m}+2j_2{D}_{j_1}^{-1}{D}_{j_2}{D}_{j_m}+3j_3{D}_{j_1}^{-2}{D}_{j_2}^2{D}_{j_m}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-(m-1)}{D}_{j_3}^{m-1}{D}_{j_m}-(n+1)D_{j_1}^{m-1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(56)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_2}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-2(k-1)}{D}_{j_3}^{k-1}{D}_{j_2}+2j_2{D}_{j_1}^{-2}{D}_{j_3}{D}_{j_2}+3j_3{D}_{j_1}^{-4}{D}_{j_3}^2{D}_{j_2}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-2(m-1)}{D}_{j_3}^{m-1}{D}_{j_2}-(n+1)D_{j_1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(57)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_3}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-2(k-1)}{D}_{j_3}^k+2j_2{D}_{j_1}^{-2}{D}_{j_3}^2+3j_3{D}_{j_1}^{-4}{D}_{j_3}^3+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-2(m-1)}{D}_{j_3}^m-(n+1)D_{j_1}^2){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(58)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_m}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-2(k-1)}{D}_{j_3}^{k-1}{D}_{j_m}+2j_2{D}_{j_1}^{-2}{D}_{j_3}{D}_{j_m}+3j_3{D}_{j_1}^{-4}{D}_{j_3}^2{D}_{j_m}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-2(m-1)}{D}_{j_3}^{m-1}{D}_{j_m}-(n+1)D_{j_1}^{m-1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(59)

$$\begin{array}{c}\hfill ⋮⋮⋮\end{array}$$

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_2}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-{(k-1)}^2}{D}_{j_m}^{k-1}{D}_{j_2}+2j_2{D}_{j_1}^{-(m-1)}{D}_{j_m}{D}_{j_2}+3j_3{D}_{j_1}^{-2(m-1)}{D}_{j_m}^2{D}_{j_2}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-{(m-1)}^2}{D}_{j_m}^{m-1}{D}_{j_2}-(n+1)D_{j_1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(60)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_3}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-{(k-1)}^2}{D}_{j_m}^{k-1}{D}_{j_3}+2j_2{D}_{j_1}^{-(m-1)}{D}_{j_m}{D}_{j_3}+3j_3{D}_{j_1}^{-2(m-1)}{D}_{j_m}^2{D}_{j_3}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-{(m-1)}^2}{D}_{j_m}^{m-1}{D}_{j_3}-(n+1)D_{j_1}^2){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(61)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_m}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{-{(k-1)}^2}{D}_{j_m}^k+2j_2{D}_{j_1}^{-(m-1)}{D}_{j_m}^2+3j_3{D}_{j_1}^{-2(m-1)}{D}_{j_m}^3+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{-{(m-1)}^2}{D}_{j_m}^m-(n+1)D_{j_1}^{m-1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(62)

Proof. 

By using the expression

$${}_{{\pounds }_{n+1}^{-}}{}_{{\pounds }_n^{+}}\left\{{}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\right\}={}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m).$$

(63)

By substituting expressions (21) and (25) into the factorization relation Equation (63), we establish the validity of the assertion Equation (54).

By utilizing expressions (22) and (25) in the factorization relation Equation (62), we establish the validity of assertion (55).

By employing expressions (23) and (25) in the factorization relation Equation (62), we verify the validity of assertion (56).

Using expressions (21)–(23) along with expression (26), we can separately prove assertions (57)–(59).

By utilizing expressions (21)–(23) in conjunction with expression (27), we can independently demonstrate the validity of assertions (60)–(62).  □

Theorem 5.

M${}_D$HAP ${}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ satisfy the following partial differential equations:

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^n{D}_{j_2}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{n-k+1}{D}_{j_2}^k+2j_2{D}_{j_1}^{n-1}{D}_{j_2}^2+3j_3{D}_{j_1}^{n-2}{D}_{j_2}^3+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{n-(m-1)}{D}_{j_2}^m-(n+1)D_{j_1}^{n+1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(64)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^n{D}_{j_3}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{n-k+1}{D}_{j_2}^{k-1}{D}_{j_3}+2j_2{D}_{j_1}^{n-1}{D}_{j_2}{D}_{j_3}+3j_3{D}_{j_1}^{n-2}{D}_{j_2}^2{D}_{j_3}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{n-(m-1)}{D}_{j_2}^{m-1}{D}_{j_3}-(n+1)D_{j_1}^{n+2}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(65)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^{2n}{D}_{j_m}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{2n-k+1}{D}_{j_2}^{k-1}{D}_{j_m}+2j_2{D}_{j_1}^{2n-1}{D}_{j_2}{D}_{j_m}+3j_3{D}_{j_1}^{2n-2}{D}_{j_2}^2{D}_{j_m}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{2n-(m-1)}{D}_{j_3}^{m-1}{D}_{j_m}-(n+1)D_{j_1}^{2n+m-1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(66)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^{2n+2}{D}_{j_2}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{2n-2k+2}{D}_{j_3}^{k-1}{D}_{j_2}+2j_2{D}_{j_1}^{2n}{D}_{j_3}{D}_{j_2}+3j_3{D}_{j_1}^{2n-2}{D}_{j_3}^2{D}_{j_2}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{2n-2m}{D}_{j_3}^{m-1}{D}_{j_2}-(n+1)D^{2n+m-1}{j}_1){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(67)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^{2n+2}{D}_{j_3}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{2n-2k+2}{D}_{j_3}^k+2j_2{D}_{j_1}^{2n}{D}_{j_3}^2+3j_3{D}_{j_1}^{2n-2}{D}_{j_3}^3+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{2n-2m}{D}_{j_3}^m-(n+1)D_{j_1}^{2n+4}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(68)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^{2n+2}{D}_{j_m}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{2n-2k+2}{D}_{j_3}^{k-1}{D}_{j_m}+2j_2{D}_{j_1}^{2n}{D}_{j_3}{D}_{j_m}+3j_3{D}_{j_1}^{2n-2}{D}_{j_3}^2{D}_{j_m}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{2n-2m}{D}_{j_3}^{m-1}{D}_{j_m}-(n+1)D_{j_1}^{2n+m+1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(69)

$$\begin{array}{c}\hfill ⋮⋮⋮\end{array}$$

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^{n^2+1}{D}_{j_2}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{n^2-k^2+2k}{D}_{j_m}^{k-1}{D}_{j_2}+2j_2{D}_{j_1}^{n^2-m}{D}_{j_m}{D}_{j_2}+3j_3{D}_{j_1}^{n^2-2m-1}{D}_{j_m}^2{D}_{j_2}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{n^2-m^2+2m}{D}_{j_m}^{m-1}{D}_{j_2}-(n+1)D_{j_1}^{n^2+2}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(70)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^{n^2+1}{D}_{j_3}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{n^2-k^2+2k}{D}_{j_m}^{k-1}{D}_{j_3}+2j_2{D}_{j_1}^{2n-m}{D}_{j_m}{D}_{j_3}+3j_3{D}_{j_1}^{2n-2m}{D}_{j_m}^2{D}_{j_3}+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{n^2-m^2+2m}{D}_{j_m}^{m-1}{D}_{j_3}-(n+1)D_{j_1}^{n^2+2}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(71)

$$\begin{array}{c}\hfill ((j_1+i_0)D_{j_1}^{n^2+2}{D}_{j_m}+\sum _{k=1}^n\frac{i_k}{k!}{D}_{j_1}^{n^2-k^2+2k}{D}_{j_m}^k+2j_2{D}_{j_1}^{n^2-m}{D}_{j_m}^2+3j_3{D}_{j_1}^{n^2-2(m-1)}{D}_{j_m}^3+\cdots \\ \hfill +m{j}_m{D}_{j_1}^{n^2-m^2+2m}{D}_{j_m}^m-(n+1)D_{j_1}^{n^2+m+1}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)=0.\end{array}$$

(72)

Proof. 

Taking derivatives of integrodifferential expressions (54) and (55) partially n times w.r.t. $j_1$, assertions (64) and (65) are proved.

Additionally, taking derivatives of integrodifferential expression (56) partially $2n$ times w.r.t. $j_1$, assertion (66) is proved.

Further, taking derivatives of integrodifferential expressions (57)–(59) partially $2n+2$ times w.r.t. $j_1$, assertions (67)–(69) are proved.

Furthermore, taking derivatives of integrodifferential expressions (60) and (61) partially $n^2+1$ times w.r.t. $j_1$, assertions (70) and (71) are proved.

Again, taking derivatives of integrodifferential expression (62) partially $n^2+2$ times w.r.t. $j_1$, assertion (72) is proved.  □

4. Applications

The Appell polynomial family is a vast collection of polynomials, which can be constructed by selecting appropriate functions $\mathfrak{A}\left(\xi \right)$.

Table 1. Expressions for certain Appell family members.

S.	Name of the	$\mathfrak{A}\left(\mathit{\xi }\right)$	Generating Expression	Series Representation
No.	Polynomials and
	Related Numbers
I.	Bernoulli	$\left(\frac{\xi }{e^{\xi }-1}\right)$	$\left(\frac{\xi }{e^{\xi }-1}\right)e^{\xi j_1}={\displaystyle \sum _{k=0}^{\infty }}{\mathfrak{B}}_k\left(j_1\right)\frac{{\xi }^k}{k!}$	${\mathfrak{B}}_k\left(j_1\right)={\displaystyle \sum _{m=0}^k}\left(\genfrac{}{}{0pt}{}{k}{m}\right){\mathfrak{B}}_m{\xi }^{k-m}$
	polynomials		$\left(\frac{\xi }{e^{\xi }-1}\right)={\displaystyle \sum _{k=0}^{\infty }}{\mathfrak{B}}_k\frac{{\xi }^k}{k!}$
	and numbers [19]		${\mathfrak{B}}_k:={\mathfrak{B}}_k\left(0\right)$
II.	Euler	$\left(\frac{2}{e^{\xi }+1}\right)$	$\left(\frac{2}{e^{\xi }+1}\right)e^{j_1\xi }={\displaystyle \sum _{k=0}^{\infty }}{\mathfrak{E}}_k\left(j_1\right)\frac{{\xi }^k}{k!}$	${\mathfrak{E}}_k\left(j_1\right)={\displaystyle \sum _{m=0}^k}\left(\genfrac{}{}{0pt}{}{k}{m}\right)\frac{{\mathfrak{E}}_m}{2^m}{\left(j_1-\frac{1}{2}\right)}^{k-m}$
	polynomials		$\frac{2e^{\xi }}{e^{2\xi }+1}={\displaystyle \sum _{k=0}^{\infty }}{\mathfrak{E}}_k\frac{{\xi }^k}{k!}$
	and numbers [20]		${\mathfrak{E}}_k:=2^k{\mathfrak{E}}_k\left(\frac{1}{2}\right)$

provides a list of various members of the Appell polynomial family, along with their corresponding names, generating functions, series definitions, and related numbers.

The Bernoulli and Euler numbers are highly significant in mathematics with numerous applications. For instance, the Bernoulli numbers, which are rational numbers, are frequently used in various mathematical formulas, such as the Bernoulli polynomials and the Euler–Maclaurin formula. They have diverse practical applications across various branches of mathematics. For instance, they find utility in number theory, numerical analysis, and combinatorics. Additionally, they are connected to other important fields, such as representation theory and algebraic geometry.

Likewise, the Euler numbers form a sequence of integers with wide-ranging applications in different mathematical disciplines, including algebraic topology, geometry, and number theory. Their significance extends to the study of elliptic curves and modular forms, with cryptography and coding theory benefiting from their applications. Moreover, these numbers are linked to the Riemann zeta function and are relevant in graph theory and automata theory.

Furthermore, the trigonometric and hyperbolic secant functions share a close association with the Euler numbers. Their Taylor series expansions incorporate the Euler numbers and their derivatives, leading to applications in numerous areas of mathematics and physics, such as signal processing and quantum field theory.

We can obtain various members of multivariable Hermite-based Bernoulli polynomials ${}_{\mathcal{Y}}{\mathfrak{B}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)$ and multivariable Hermite-based Euler polynomials ${}_{\mathcal{Y}}{\mathfrak{E}}_n^{\left[m\right]}(j_1,j_2,j_3,$$\cdots ,j_m)$ by taking Bernoulli and Euler polynomials as members of the Appell family. The generating expressions for these polynomials are provided as follows:

$$\frac{\xi }{(e^{\xi }-1)}exp(j_1\xi +j_2{\xi }^2+\cdots +j_m{\xi }^m=\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{B}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\frac{{\xi }^n}{n!}$$

(73)

and

$$\frac{2}{(e^{\xi }+1)}exp(j_1\xi +j_2{\xi }^2+\cdots +j_m{\xi }^m=\sum _{n=0}^{\infty }{}_{\mathcal{Y}}{\mathfrak{E}}_n^{\left[m\right]}(j_1,j_2,j_3,\cdots ,j_m)\frac{{\xi }^n}{n!}$$

(74)

respectively, with the radius of convergence as $2\pi$ and $\pi$, respectively, by using D’Almbert’s ratio test.

As a result, we can extend the analysis to derive analogous properties for the multivariable Hermite-based Bernoulli and Euler polynomials. Recurrence relations for multivariable Hermite-based Bernoulli and Euler polynomials are obtained in the first step of our analysis:

Corollary 1.

The multivariable Hermite-based Bernoulli ${}_{\mathcal{Y}}{\mathfrak{B}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)$ satisfy the following recurrence relation:

$$\begin{array}{c}{}_{\mathcal{Y}}{\mathfrak{B}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=(j_1-\frac{1}{2}){}_{\mathcal{Y}}{\mathfrak{A}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)-\frac{1}{n+1}\sum _{k=2}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{k}\right)B_k\hfill \\ \times {}_{\mathcal{Y}}{\mathfrak{B}}_{n-k+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+j_2{}_{\mathcal{Y}}{\mathfrak{B}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+3n(n-1)j_3{}_{\mathcal{Y}}{\mathfrak{B}}_{n-2}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\\ \hfill +\cdots +n(n-1)(n-2)\cdots (n-m+1)j_m{}_{\mathcal{Y}}{\mathfrak{B}}_{n-m}^{\left[m\right]}(j_1,j_2,\cdots ,j_m),\end{array}$$

(75)

where $B_k$ denotes the Bernoulli number of order k.

Corollary 2.

The multivariable Hermite-based Euler ${}_{\mathcal{Y}}{\mathfrak{E}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)$ satisfy the following recurrence relation:

$$\begin{array}{c}{}_{\mathcal{Y}}{\mathfrak{E}}_{n+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)=(j_1-\frac{1}{2}){}_{\mathcal{Y}}{\mathfrak{E}}_n^{\left[m\right]}(j_1,j_2,\cdots ,j_m)-\frac{1}{n+1}\sum _{k=2}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{k}\right)E_k\hfill \\ \times {}_{\mathcal{Y}}{\mathfrak{E}}_{n-k+1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+j_2{}_{\mathcal{Y}}{\mathfrak{E}}_{n-1}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)+3n(n-1)j_3{}_{\mathcal{Y}}{\mathfrak{E}}_{n-2}^{\left[m\right]}(j_1,j_2,\cdots ,j_m)\\ \hfill +\cdots +n(n-1)(n-2)\cdots (n-m+1)j_m{}_{\mathcal{Y}}{\mathfrak{E}}_{n-m}^{\left[m\right]}(j_1,j_2,\cdots ,j_m),\end{array}$$

(76)

where $E_k$ denotes the Euler number of order k.

Further, continuing in a similar fashion, we can establish the shift operators and families of differential equations for these polynomials by using simple calculations. The results are omitted here.

The fractional operators such as Euler integral of the IInd kind may be taken as future observation to derive the extended form of these special polynomial families.

5. Conclusions

In this study, a novel family of hybrid multivariable polynomials is introduced by convolving Hermite and Appell polynomials. The properties of these polynomials are investigated in detail. To be specific, we derive a recurrence relation and a sequence of shift operators that are satisfied by the multivariable Hermite–Appell polynomials. Furthermore, we establish that these polynomials fulfill a differential equation and a sequence of integrodifferential and partial differential equations. Additionally, we look at numerous Appell family members and establish the generating and recurring relations that correspond to them. Overall, by proposing a new family of polynomials and examining their characteristics, this paper makes a contribution to the subject of polynomial theory.

Additionally, with additional research and observations, it could be able to investigate new characteristics of the aforementioned polynomials. Creating extended and generalized forms, symmetric identities, Volterra integral equations, and other traits could be included in this. However, difficulties with determinant forms and summation equations might arise when dealing with fresh data.

In the future, more research and observations may make it feasible to develop new characteristics of the aforementioned polynomials. These qualities could include symmetric identities, extended and generalized forms utilizing fractional operators, and other traits. However, difficulties with determinant forms and summation equations might arise when dealing with fresh data.