A Study on Generalized Degenerate Form of 2D Appell Polynomials via Fractional Operators

Abstract

This paper investigates the significance of generating expressions, operational principles, and defining characteristics in the study and development of special polynomials. The focus is on a novel generalized family of degenerate 2D Appell polynomials, which were defined using a fractional operator. Motivated by inquiries into degenerate 2D bivariate Appell polynomials, this research reveals that well-known 2D Appell polynomials and simple Appell polynomials can be regarded as specific instances within this new family for certain values. This paper presents the operational rule, generating relation, determinant form, and recurrence relations for this generalized family. Furthermore, it explores the practical applications of these degenerate 2D Appell polynomials and establishes their connections with equivalent results for the generalized family of degenerate 2D Bernoulli, Euler, and Genocchi polynomials.

1. Introduction

Special polynomials form a distinct group of mathematical functions known for their exceptional and valuable properties, rendering them indispensable and captivating across diverse domains such as mathematics, physics, engineering, and other scientific disciplines. What sets these polynomials apart from regular ones is their distinct attributes, including specific generating expressions, operational rules, and other defining characteristics. Special polynomials and fractional operators have a noteworthy relationship within the realm of mathematics. Fractional operators introduce a new dimension for ordinary differentiation and integration by extending these operations to non-integer orders, which are often represented by fractional exponents. Special polynomials, on the other hand, are a class of mathematical functions with unique properties and generated expressions. The combination of special polynomials with fractional operators leads to the development of more sophisticated and versatile mathematical tools. By applying fractional operators to special polynomials, researchers can create new families of fractional special polynomials, which can offer enhanced capabilities in solving complex mathematical problems and the modeling of various phenomena. This synergy has proven valuable in numerous fields, including mathematical physics, signal processing, and control theory. Fractional special polynomials provide a powerful framework for addressing problems involving fractional calculus and fractional differential equations, offering insights and solutions beyond what traditional integer-order polynomials can achieve. As a result, the study of special polynomials in conjunction with fractional operators continues to be an active and fruitful area of research in mathematics and its applications.

The idea of fractional calculus, which includes integrating non-integral orders, has a long history. Its roots may be found in the beginnings of differential calculus when, in the late 17th century, the renowned mathematician and philosopher Leibniz—who competed with Newton—offered the idea and possibility of a fractional derivative of order $1/2$. But it was not until Liouville produced a devoted effort on this matter that a careful and full investigation of this subject was made, thus producing accurate and rigorous research. Both differential and integral equations can be solved with the use of integral transforms. As seen in the works of Oldham and Widder [1,2], fractional operators have long piqued the curiosity of mathematicians and engineers. According to the academic publications of [1,2], the use of integral transforms to solve fractional derivatives may be traced back to the contributions of Riemann and Liouville. The synergistic utilization of integral transformations and specialized polynomials offers a powerful approach in dealing with fractional derivatives. This strategy has been acknowledged as a potent instrument; see, for instance [3,4], where the authors of those studies underline the importance of this combination strategy and give further information about the real-world uses and theoretical developments made possible by the use of integral transformations and specific polynomials when dealing with fractional derivatives. The advantages of this technique have been thoroughly investigated by researchers and practitioners, thus contributing to a greater knowledge of fractional calculus and its wide variety of applications.

A reliable and efficient method for managing fractional derivatives is provided by the combination of integral transformations and special polynomials. This strategy has drawn a lot of interest and is recognized as a potent instrument in the industry. Researchers and practitioners can develop effective techniques for analyzing and resolving fractional differential equations by combining integral transforms, such as Laplace or Fourier transforms, with unique polynomials, such as Hermite, Laguerre, or Chebyshev polynomials. These methods have proven useful in a variety of industries, including engineering, banking, and signal processing, as well as in physics.

In [3], Dattoli and coauthors explored the potential of using integral transforms in a wider context. In their research, they examine using Euler’s integral to extend the use of integral transforms beyond their typical limitations. A comprehensive foundation for increasing the applicability and efficiency of integral transformations in numerous domains is provided by Euler’s integral:

$${\sigma }^{-\mu }=\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\sigma u}{u}^{\mu -1}du,$$

(1)

where $\sigma$ and $\mu$ are complex numbers with positive real parts and satisfy $min\left\{\mathrm{Re}\right(\mu ),\mathrm{Re}(\sigma \left)\right\}>0.$ Researchers may tackle a larger range of issues by bringing Euler’s integral within the framework of integral transformations, enabling for the study and solution of challenging mathematical equations found in a variety of domains. This broader framework allows for novel perspectives in terms of investigating fractional derivatives and the implications that they have, thus producing original ideas and approaches to tackling problems. This study shows the potential for further advancement in this area, and provides both researchers and practitioners with a useful tool for solving difficult issues involving fractional derivatives in a broader context.

Further, in [3], it is evident that, for derivatives of a first and second order, the listed axioms hold:

$$\begin{array}{cc}\hfill {\left(\beta -\frac{\partial }{\partial p}\right)}^{-\mu }g\left(p\right)=& \frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\beta t}{t}^{\mu -1}{e}^{u\frac{\partial }{\partial p}}g\left(p\right)du=\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\beta t}{u}^{\mu -1}g(p+u)du\end{array}$$

(2)

$${\left(\beta -\frac{{\partial }^2}{\partial p^2}\right)}^{-\mu }g\left(p\right)=\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\beta u}{u}^{\mu -1}{e}^{u\frac{{\partial }^2}{\partial p^2}}g\left(p\right)du,$$

(3)

A synergistic combination of exponential operators and suitable integral representations may be used to efficiently address fractional operators. Researchers and practitioners can efficiently handle fractional operators in a streamlined manner by making use of the characteristics of exponential operators and by choosing appropriate integral representations. This method makes it possible to explore cutting-edge mathematical ideas and makes it easier to accurately and quickly analyze fractional derivatives. A strong foundation for handling fractional operators is provided by the use of exponential operators and specialized integral representations, which leads to improved approaches and solutions across a range of mathematical and scientific areas.

The Appell polynomials [5] find numerous applications in diverse fields of mathematics and physics, including algebraic geometry, differential equations, and quantum mechanics. They demonstrate significant connections with other special function families, such as hypergeometric functions and Jacobi polynomials. When two Appell polynomials are composed, they yield another Appell polynomial, thereby forming an abelian group under composition. This group property arises due to the Appell polynomials’ differential equation being a specific case of the Heun equation, known for its Galois group being an abelian extension of the differential field generated by the equation’s solutions.

Precisely speaking, the sequences of Appell polynomials form an abelian group when composed, which is where the constant polynomial sequence serves as the identity element. The commutativity of this group arises from the inherent symmetry of the Appell polynomials. The group property of the Appell polynomials carries significant implications across various fields of mathematics and physics, particularly in the realms of differential equation theory and integrable systems research. By leveraging this group property, one can derive recursion relations for the coefficients of Appell polynomials, which prove valuable in computing special polynomial values. Furthermore, this property allows for the construction of novel families of Appell polynomials by composing existing ones, leading to the exploration and discovery of new intriguing mathematical structures.

Appell [5] in the 19th century presented a family of polynomials known as the Appell polynomial family. It is given by ${\mathbb{R}}_k\left(p\right),p\in \mathbb{C}$, which satisfies the following differential equation:

$$\frac{d}{dp}{\mathbb{R}}_k\left(p\right)=k{\mathbb{R}}_{k-1}\left(p\right),k\in {\mathbb{N}}_0$$

(4)

and generates the following relation expression:

$$\mathbb{R}\left(u\right)e^{pu}=\sum _{k=0}^{\infty }{\mathbb{R}}_k\left(p\right)\frac{u^k}{k!},$$

(5)

where $\mathbb{R}\left(u\right)$ is represented by

$$\mathbb{R}\left(u\right)=\sum _{k=0}^{\infty }{\mathbb{R}}_k\frac{u^k}{k!},$$

(6)

which is a formal power series with complex coefficients ${\mathbb{R}}_k$, $k=0,1,2,\cdots$, ${\mathbb{R}}_0\ne 0$.

These polynomials were introduced by the French mathematician Paul Appell in his research on elliptic functions. The generating relation (5) offers a method through which to represent the exponential function $e^{pu}$ as an infinite sum of the polynomials ${\mathbb{R}}_k\left(p\right)$, which are multiplied by powers of u. This relation proves useful in simplifying certain integrals and in evaluating specific functions. In mathematical physics, the Appell polynomials find numerous applications, particularly in the study of quantum mechanics, electromagnetism, and fluid dynamics. They play a significant role in these fields, enhancing our understanding, and facilitating calculations in various contexts.

Recently, the 2D Appell polynomials [6] have been represented by the following generating relation:

$$\sum _{l=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_l^{\left(j\right)}(p,q)\frac{u^l}{l!}=\mathbb{R}\left(u\right)exp(pt+q{t}^j),p,q\in \mathbb{C}$$

(7)

where $\mathbb{R}\left(u\right)$ is given by (6), and acts as a solution of the following heat equation:

$$\begin{array}{cc}\frac{\partial }{\partial q}\{{}_{\mathbb{G}}{\mathbb{R}}_l^{\left(j\right)}(p,q)\}& =\frac{{\partial }^j}{\partial p^j}\{{}_{\mathbb{G}}{\mathbb{R}}_l^{\left(j\right)}(p,q)\}\\ \mathrm{and}\\ {}_{\mathbb{G}}{\mathbb{R}}_l^{\left(j\right)}(p,0)& =R_l\left(p\right).\end{array}$$

(8)

These polynomials are significant in numerous branches of mathematics, such as enumerative combinatorics, algebraic combinatorics, and applied mathematics. These polynomials are employed to solve certain ordinary differential equations in approximation theory and physics. Hermite polynomials are a noteworthy subgroup of these polynomial sequences. The group of orthogonal polynomials known as Hermite polynomials was first developed by Hermite himself [7]. Numerous mathematical, physics, engineering, and computer science problems may be solved using these polynomials. Harmonic oscillator wave functions are described by Hermite polynomials. The quantum theory of light and research on the hydrogen atom both heavily rely on them. Additionally, the probability distribution function of the kinetic energy of gas molecule molecules is derived using these polynomials. They are also employed to examine the energy distribution of a harmonic oscillator.

Many writers are interested in developing and discovering various properties of degenerate special polynomials (for instance [8,9,10,11,12,13]). Wani and colleagues recently produced several doped polynomials of a certain type, as well as identified their different traits and behaviors, which are crucial from an engineering perspective (for example [14,15,16,17,18]). Examples of these significant properties involve determinant forms, operational formalism, approximation properties, poly forms, degenerate forms, summation formulas, approximation quality, explicit and implicit formulae, generating expressions, etc.

The study of degenerate forms of special functions is crucial for understanding the mathematical foundations of many physical phenomena. The degenerate polynomials are used to describe the behavior of quantum systems, such as the quantum harmonic oscillator. These functions have found wide-ranging applications in various fields of science and engineering, and their significance has been acknowledged in both pure mathematics and practical contexts. Future discoveries and breakthroughs are anticipated as the theory of special functions continues to progress (see for instance [9,10]).

Recently, Shahid et al. [19] introduced 2-variable degenerate Appell polynomials by generating the following expression:

$$\sum _{s=0}^{\infty }{\mathbb{G}}_s^{\left[j\right]}(p,q;\lambda )\frac{u^s}{s!}=\mathbb{R}\left(u\right){(1+\lambda )}^{(\frac{pu+q{u}^j}{\lambda })},\lambda \in \mathbb{C}$$

(9)

and the operational representation as follows:

$$exp\left(q\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right){\mathbb{R}}_s(p;\lambda )={\mathbb{G}}_s^{\left[j\right]}(p,q;\lambda ).$$

(10)

It is evident that degenerate 2D bivariate Appell polynomials reduce to degenerate Appell polynomials [20], which are provided by $v\to 0$ as follows:

$$\sum _{s=0}^{\infty }{\mathbb{G}}_m^{\left[j\right]}(p,0;\lambda )\frac{u^s}{s!}=\mathbb{R}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}=\sum _{s=0}^{\infty }{\mathbb{R}}_s(p;\lambda )\frac{u^s}{s!}.$$

(11)

Also, if ${(1+\lambda )}^{\frac{pu}{\lambda }}\to exp\left(pu\right)$, then Expression (11) reduces to (5).

Moreover, as ${(1+\lambda )}^{\frac{pu}{\lambda }}\to exp\left(pu\right)$ and ${(1+\lambda )}^{\frac{q{u}^j}{\lambda }}\to exp\left(q{u}^j\right)$, then (9) reduces to (7).

In the area of mathematical analysis, fractional calculus is one of the most swiftly developing areas. Applications in a wide range of fields, including biology, physics, electrochemistry, economics, probability theory, and statistics, are possible. Fractional operators provide a more precise depiction of intricate systems that cannot be adequately represented by derivatives of an integer order. Consequently, they find extensive use across various disciplines, including multiple branches of mathematics, physics [21] engineering [22], and finance [23]. For instance, fractional operators are employed to describe the characteristics of viscoelastic materials, biological systems, and electrical networks [24]. Moreover, in the realm of electromagnetics, fractional operators play a crucial role in elucidating the behavior of electromagnetic waves within media that possess fractional-order dielectric and magnetic properties [25]. Further applications of fractional calculus can be observed in [26].

The research conducted in this paper draws inspiration and motivation from the groundbreaking work of Dattoli and colleagues [3] as well as from Wani et al. [19], who highlighted the immense significance of fractional operators. In this study, we construct the generalized form of a degenerate special polynomial family by utilizing the fractional operator known as Euler’s integral (1). We introduce the generalized degenerate 2D Appell polynomial family, denoted as ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, through the generating expression given in the form of the following definition:

Definition 1. 

$$\frac{\mathbb{R}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{\left(q{u}^j\right)}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}.$$

(12)

The application of these degenerate special polynomials extends to image processing and computer vision, thereby facilitating tasks such as image enhancement and feature extraction. Additionally, they find utility in financial mathematics, providing models to analyze the behavior of stock prices, interest rates, and other financial variables.

The purpose of this article is to carry out a comprehensive analysis of the properties of the generalized versions of degenerate special polynomials that are associated with 2D Appell polynomials. This research makes extensive use of integral transformations and operational ideas. This paper’s major contributions are included in Section 2 and Section 3, which follow an extensive introduction that highlights significant data that were previously known. In Section 2, the novelty of the discovery is demonstrated by using fractional operators to build a generalized version of degenerate 2D Appell polynomials in the form of operational formulae and generating functions. Further study of these established polynomials led to the development of a summation formula and recurrence relations. In Section 3, the determinant form for these polynomials and their special cases are deduced. This article explores the potential applications for exceptional outcomes in Section 4 along with the generalized degenerate 2D Appell polynomials; furthermore, these applications also contain comparable results for degenerate 2D-Bernoulli and 2D-Euler polynomials. By looking at various applications, this article demonstrates the broad applicability and relevance of the discovered results in several disciplines of mathematics and beyond.

2. Generalized Degenerate 2D Appell Polynomials

This section examines the newly discovered class of generalized degenerate 2D Appell polynomials. Moreover, these polynomials are also produced with a number of properties. First, we derive the operational connection of the generalized degenerate 2D Appell polynomials by demonstrating the following result:

Theorem 1. 

The generalized degenerate 2D Appell polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ satisfy the following operational connection:

$${\left(\beta -q\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right)}^{-\mu }{\mathbb{R}}_s(p;\lambda )={}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ).$$

(13)

Proof. 

On substituting $\sigma$ by $\left(\beta -\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right)$ in Expression (1) of Euler’s integral, and then operating the resultant equation on ${\mathbb{R}}_s(p;\lambda )$, we find

$${\left(\beta -\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right)}^{-\mu }{\mathbb{R}}_s(p;\lambda )=\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\beta u}{u}^{\mu -1}exp\left(qu\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial q^j}\right){\mathbb{R}}_s(p;\lambda )du.$$

(14)

It is clear from Equation (10) that the following outcome is attained from the following (14):

$${\left(\beta -\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right)}^{-\mu }{\mathbb{R}}_s(p;\lambda )=\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\beta u}{u}^{\mu -1}{}_{\mathbb{G}}{\mathbb{R}}_s^{\left[j\right]}(p,qu;\lambda )du.$$

(15)

By applying the transformation described on the right-hand side of Equation (15), a novel set of polynomials is introduced. These polynomials are denoted by ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ;\beta )$, and are referred to as the generalized degenerate 2D Appell polynomials. Thus, we establish the following relationship:

$$\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\beta u}{u}^{\mu -1}{}_{\mathbb{G}}{\mathbb{R}}_s^{\left[j\right]}(p,qu;\lambda )du={}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ;\beta ).$$

(16)

Therefore, on consideration of Expressions (15) and (16), Assertion (13) is established. □

Remark 1. 

The generalized degenerate 2D Appell polynomials can be reduced to the generalized degenerate 2-variable Hermite–Appell polynomials if $j=2$. As a result, the following operational connection is established by replacing $j=2$ in the left-hand side of Equation (13) and representing the resulting generalized degenerate 2-variable Hermite–Appell polynomials on the right-hand side as ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )$, which are expressed as follows:

$${\left(\beta -q\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^2}{\partial p^2}\right)}^{-\mu }{\mathbb{R}}_s(p;\lambda )={}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[2\right]}(p,q;\lambda ,\beta ).$$

Remark 2. 

The generalized degenerate 2D Appell polynomials can be reduced to the generalized 2D Appell polynomials if $\lambda \to 0$. As a result, the following operational connection is established by replacing $\lambda \to 0$ in the left-hand side of Equation (13) and representing the resulting generalized Appell polynomials on the right-hand side as ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }(p;q;\beta )$, which are expressed as follows:

$${\left(\beta -q\frac{{\partial }^j}{\partial p^j}\right)}^{-\mu }{\mathbb{R}}_s\left(p\right)={}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\beta ).$$

Theorem 2. 

For the generalized degenerate 2D Appell polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ;\beta )$, the listed generating expression holds true:

$$\frac{\mathbb{R}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{q{u}^j}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}.$$

(17)

Proof. 

On multiplication of Expression (16) on b/s with $\frac{u^s}{s!}$, and by subsequently summing over all values of s, it follows that

$$\sum _{s=0}^{\infty }\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }{e}^{-\beta u}{u}^{\mu -1}{}_{\mathbb{G}}{\mathbb{R}}_s^{\left[j\right]}(p,qu;\lambda )du\frac{u^s}{s!}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ;\beta )\frac{u^s}{s!}.$$

(18)

Thus, in view of Expression (9) on the l.h.s. of the previous equation, we find

$$\frac{1}{\Gamma \left(\mu \right)}{\int }_0^{\infty }\mathbb{R}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}{e}^{-\left(\beta -\frac{q{u}^j}{\lambda }log(1+\lambda )\right)u}{u}^{\mu -1}du=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ;\beta )\frac{u^s}{s!}.$$

(19)

on consideration of the integral Expression (1), Assertion (17) is deduced. □

Remark 3. 

The generalized degenerate 2D Appell polynomials can be reduced to the generalized degenerate 2-variable Hermite–Appell polynomials if $j=2$. As a result, the following generating expression is established by replacing $j=2$ on the left-hand side of Equation (13) and by representing the resulting generalized degenerate 2-variable Hermite–Appell polynomials on the right-hand side as ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )$, which are expressed as follows:

$$\frac{\mathbb{R}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{q{u}^2}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}.$$

(20)

Remark 4. 

The generalized degenerate 2D Appell polynomials can be reduced to the generalized 2D Appell polynomials if $\lambda \to 0$. As a result, the following generating expression is established by replacing $\lambda \to 0$ on the left-hand side of Equation (13) and by representing the resulting generalized Appell polynomials on the right-hand side as ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }(p;q;\beta )$, which are expressed as follows:

$$\frac{\mathbb{R}\left(u\right)e^{pu}}{{\left[\beta -q{u}^j\right]}^{\mu }}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\beta )\frac{u^s}{s!}.$$

(21)

Remark 5. 

The generalized degenerate 2D Appell polynomials can be reduced to the generalized degenerate Gould-Hopper polynomials if $\mathbb{R}\left(u\right)=1$. As a result, the following generating expression is established by taking $\mathbb{R}\left(u\right)=1$ on the left-hand side of Equation (13) and by representing the resulting generalized degenerate Gould-Hopper polynomials on the right-hand side as ${\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, which are expressed as follows:

$$\frac{{(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{\left(q{u}^j\right)}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{s=0}^{\infty }{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}.$$

(22)

Next, in view of the above Expression (22), the following generated expression holds true for the generalized degenerate 2D Appell polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ in terms of the generalized degenerate degenerate Gould-Hopper polynomials ${\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, which are expressed as follows:

Theorem 3. 

For the generalized degenerate 2D Appell polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, the succeeding generating expression holds true:

$${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )=\sum _{k=0}^s\left(\genfrac{}{}{0pt}{}{s}{k}\right){\mathbb{R}}_k{\mathbb{G}}_{s-k,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ).$$

(23)

Proof. 

Expression (12), in terms of Expressions (6) and (22), can be written as

$$\frac{\mathbb{R}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{\left(q{u}^j\right)}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{k=0}^{\infty }{\mathbb{R}}_k\frac{u^k}{k!}\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}.$$

(24)

Therefore, in view of the Cauchy product rule [27] (p. 85, Equation (5)), Assertion (23) is established. □

Next, by taking into account the generating relation of generalized degenerate 2D Appell polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, the recurrence relations for these polynomials can be deduced. Equations that recursively specify the terms of the sequence or array can be used to build recurrence relations for a sequence or multidimensional array of values. These relationships enable us to express each subsequent term in relation to the preceding ones. When we wish to construct the values of a sequence or array iteratively, beginning with one or more initial terms, recurrence relations come in extremely handy. By taking derivatives with respect to p, q, and the $\beta$ for generating Expression (17), the succeeding recurrence relations of polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ are deduced as follows:

$$\begin{array}{ccc}\hfill \frac{\lambda }{log(1+\lambda )}\frac{\partial }{\partial p}\left({}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\right)& =& s{}_{\mathbb{G}}{\mathbb{R}}_{s-1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ),\hfill \\ \hfill \frac{\lambda }{log(1+\lambda )}\frac{\partial }{\partial q}\left({}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\right)& =& \mu s(s-1)\cdots (s-j+1){}_{\mathbb{G}}{\mathbb{R}}_{s-j,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ),\hfill \\ \hfill \frac{\partial }{\partial \beta }\left({}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\right)& =& -\mu {}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ).\hfill \end{array}$$

(25)

In consideration of the above relations, the succeeding expression holds true:

$$\begin{array}{c}\hfill {\left(\frac{\lambda }{log(1+\lambda )}\right)}^{j-1}\frac{\partial }{\partial q}\left({}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\right)=-\frac{{\partial }^{j+1}}{\partial p^j\partial \beta }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ).\end{array}$$

(26)

3. Determinant Forms

Determinants play a significant role in many branches of mathematics and its applications. Determinants are used to create the orthogonality connections between the functions, which are necessary for many applications in mathematics and physics. They also help in normalizing the functions, ensuring that their squared magnitudes integrate into unity. They are utilized to create generating functions for special functions. They act as a powerful method for obtaining difference equations, recursion relations, and other properties of special functions. Determinants commonly appear in the generating functions as coefficients.

Here, we establish the determinant form for the generalized degenerate 2D Appell polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, which are given by Expression (17) in the form of the following result:

Theorem 4. 

The generalized degenerate 2D Appell polynomials ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ give rise to the determinant of the following form:

$${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )=\frac{{(-1)}^s}{{\left({\gamma }_0\right)}^{s+1}}\left|\begin{array}{cccccc}1& {\mathbb{G}}_{1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{G}}_{2,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& \cdots & {\mathbb{G}}_{s-1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\\ \\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{s-1}& {\gamma }_s\\ \\ 0& {\gamma }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\gamma }_1& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{1}\right){\gamma }_{s-2}& \left(\genfrac{}{}{0pt}{}{s}{1}\right){\gamma }_{s-1}\\ \\ 0& 0& {\gamma }_0& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{2}\right){\gamma }_{s-3}& \left(\genfrac{}{}{0pt}{}{s}{2}\right){\gamma }_{s-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& \left(\genfrac{}{}{0pt}{}{s}{s-1}\right){\gamma }_1\end{array}\right|,$$

(27)

where γ_s, s = 0, 1, ⋯ are the coefficients of Maclaurin’s series$\frac{1}{\mathbb{R}\left(u\right)}$.

Proof. 

In multiplying both sides of Equation (17) by $\frac{1}{\mathbb{R}\left(u\right)}={\displaystyle \sum _{k=0}^{\infty }}{\gamma }_k\frac{u^k}{k!}$, we find

$$\sum _{s=0}^{\infty }{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}=\sum _{k=0}^{\infty }{\gamma }_k\frac{u^k}{k!}\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}.$$

(28)

By using the Cauchy product rule in Equation (28), we have

$${\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )=\sum _{k=0}^s\left(\genfrac{}{}{0pt}{}{s}{k}\right){\gamma }_k{}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ).$$

(29)

The system of $s-$ equations with unknown ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ),s=0,1,2,\cdots$ is produced as a result of this equality. Cramer’s rule is used to solve this system, and by using the fact that the denominator is the determinant of the lower triangular matrix with determinant ${\left({\gamma }_0\right)}^{s+1}$, as well as by taking the transpose of the numerator, then replacing the $i-th$ row by $(i+1)-th$ position for $i=1,2,\cdots s-1$ gives the desired result. □

Remark 6. 

The generalized degenerate 2D Appell polynomials can be reduced to the generalized degenerate 2-variable Hermite–Appell polynomials if $j=2$. As a result, the following determinant representation is established by replacing $j=2$ in Expression (27) and by representing the resulting generalized degenerate 2-variable Hermite–Appell polynomials on the right-hand side as ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )$, which are expressed as follows:

$${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )=\frac{{(-1)}^s}{{\left({\gamma }_0\right)}^{s+1}}\left|\begin{array}{cccccc}1& {\mathbb{R}}_{1,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )& {\mathbb{R}}_{2,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )& \cdots & {\mathbb{R}}_{s-1,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )& {\mathbb{R}}_{s,\mu }^{\left[2\right]}(p,q;\lambda ,\beta )\\ \\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{s-1}& {\gamma }_s\\ \\ 0& {\gamma }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\gamma }_1& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{1}\right){\gamma }_{s-2}& \left(\genfrac{}{}{0pt}{}{s}{1}\right){\gamma }_{s-1}\\ \\ 0& 0& {\gamma }_0& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{2}\right){\gamma }_{s-3}& \left(\genfrac{}{}{0pt}{}{s}{2}\right){\gamma }_{s-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& \left(\genfrac{}{}{0pt}{}{s}{s-1}\right){\gamma }_1\end{array}\right|,$$

Remark 7. 

The generalized degenerate 2D Appell polynomials can be reduced to the generalized 2D Appell polynomials if $\lambda \to 0$. As a result, the following determinant representation is established by replacing $\lambda \to 0$ in Expression (27) and by representing the resulting generalized Appell polynomials on the right-hand side as ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }(p;q;\beta )$, which are expressed as follows:

$${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\beta )=\frac{{(-1)}^s}{{\left({\gamma }_0\right)}^{s+1}}\left|\begin{array}{cccccc}1& {\mathbb{G}}_{1,\mu }^{\left[j\right]}(p,q;\beta )& {\mathbb{G}}_{2,\mu }^{\left[j\right]}(p,q;\beta )& \cdots & {\mathbb{G}}_{s-1,\mu }^{\left[j\right]}(p,q;\beta )& {\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\beta )\\ \\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{s-1}& {\gamma }_s\\ \\ 0& {\gamma }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\gamma }_1& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{1}\right){\gamma }_{s-2}& \left(\genfrac{}{}{0pt}{}{s}{1}\right){\gamma }_{s-1}\\ \\ 0& 0& {\gamma }_0& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{2}\right){\gamma }_{s-3}& \left(\genfrac{}{}{0pt}{}{s}{2}\right){\gamma }_{s-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& \left(\genfrac{}{}{0pt}{}{s}{s-1}\right){\gamma }_1\end{array}\right|.$$

4. Applications

The Appell polynomial family is a vast collection of polynomials that can be constructed by selecting the appropriate functions $\mathbb{R}\left(u\right)$.

Table 1. Expressions for certain Appell family members.

S. No.	Name of The Polynomials and Related Numbers	$\mathbb{R}\left(\mathit{u}\right)$	Generating Expression	Series Representation
I.	Bernoulli	$\left(\frac{u}{e^u-1}\right)$	$\left(\frac{u}{e^u-1}\right)e^{u{j}_1}={\displaystyle \sum _{k=0}^{\infty }}{\mathbb{B}}_k\left(j_1\right)\frac{u^k}{k!}$	${\mathbb{B}}_k\left(j_1\right)={\displaystyle \sum _{m=0}^k}\left(\genfrac{}{}{0pt}{}{k}{m}\right){\mathbb{B}}_m{u}^{k-m}$
	polynomials		$\left(\frac{u}{e^u-1}\right)={\displaystyle \sum _{k=0}^{\infty }}{\mathbb{B}}_k\frac{u^k}{k!}$
	and numbers [28]		${\mathbb{B}}_k:={\mathbb{B}}_k\left(0\right)$
II.	Euler	$\left(\frac{2}{e^u+1}\right)$	$\left(\frac{2}{e^u+1}\right)e^{j_{1}u}={\displaystyle \sum _{k=0}^{\infty }}{\mathbb{E}}_k\left(j_1\right)\frac{u^k}{k!}$	$\begin{array}{c}{\mathbb{E}}_k\left(j_1\right)=\hfill \\ {\displaystyle \sum _{m=0}^k}\left(\genfrac{}{}{0pt}{}{k}{m}\right)\frac{{\mathbb{E}}_m}{2^m}{\left(j_1-\frac{1}{2}\right)}^{k-m}\hfill \end{array}$
	polynomials		$\frac{2e^u}{e^{2u}+1}={\displaystyle \sum _{k=0}^{\infty }}{\mathbb{E}}_k\frac{u^s}{s!}$
	and numbers [28]		${\mathbb{E}}_k:=2^k{\mathbb{E}}_k\left(\frac{1}{2}\right)$
III.	Genocchi	$\left(\frac{2u}{e^u+1}\right)$	$\left(\frac{2u}{e^u+1}\right)e^{j_{1}u}={\displaystyle \sum _{k=0}^{\infty }}{\mathbb{G}}_k\left(j_1\right)\frac{u^k}{k!}$	${\mathbb{G}}_k\left(j_1\right)={\displaystyle \sum _{m=0}^k}\left(\genfrac{}{}{0pt}{}{k}{m}\right){\mathbb{G}}_m{j_1}^{k-m}$
	polynomials		$\frac{2u}{e^u+1}={\displaystyle \sum _{k=0}^{\infty }}{\mathbb{G}}_k\frac{u^s}{s!}$
	and numbers [29]		${\mathbb{G}}_k:={\mathbb{G}}_k\left(0\right)$

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

The Bernoulli, Euler, and Genocchi numbers hold immense significance in mathematics, thus finding numerous applications in various fields. For example, the Bernoulli numbers are frequently employed in diverse mathematical formulas, such as the Bernoulli polynomials and the Euler–Maclaurin formula. They have practical uses in number theory, numerical analysis, and combinatorics, and they are also closely related to representation theory and algebraic geometry.

Similarly, the integer sequence of Euler numbers plays a vital role in different mathematical domains, including algebraic topology, geometry, and number theory. These numbers are essential in the study of elliptic curves and the theory of modular forms, which find applications in cryptography and coding theory.

On the other hand, the Genocchi numbers, which also form an integer sequence, are applied to various combinatorial problems, including the counting of labeled rooted trees and up–down sequences. They are linked to the Riemann zeta function and prove useful in graph theory and automata theory.

Moreover, the trigonometric and hyperbolic secant functions are closely connected to the Euler numbers. The Taylor series expansions of these functions involve Euler numbers and their derivatives, making them valuable in many areas of mathematics and physics, such as signal processing and quantum field theory.

Therefore, by replacing $\mathbb{R}\left(u\right)$ with $\mathbb{B}\left(u\right)$, $\mathbb{E}\left(u\right)$, and $\mathbb{G}\left(u\right)$ in Expressions (13) and (17), the listed expression’s for the operational connection and generating function for the generalized degenerate 2D Bernoulli, Euler, and Genocchi polynomials are established as follows:

Corollary 1. 

The generalized degenerate 2D Bernoulli ${}_{\mathbb{G}}{\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, Euler ${}_{\mathbb{G}}{\mathbb{E}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, and Genocchi ${}_{\mathbb{G}}{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ polynomials satisfy the following operational connection:

$$\begin{array}{c}{\left(\beta -q\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right)}^{-\mu }{\mathbb{B}}_s(p;\lambda )={}_{\mathbb{G}}{\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ),\hfill \\ {\left(\beta -q\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right)}^{-\mu }{\mathbb{E}}_s(p;\lambda )={}_{\mathbb{G}}{\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ),\hfill \\ {\left(\beta -q\frac{\lambda }{log(1+\lambda )}\frac{{\partial }^j}{\partial p^j}\right)}^{-\mu }{\mathbb{B}}_s(p;\lambda )={}_{\mathbb{G}}{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta ).\hfill \end{array}$$

(30)

Corollary 2. 

The generalized degenerate 2D Bernoulli ${}_{\mathbb{G}}{\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, Euler ${}_{\mathbb{G}}{\mathbb{E}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$, and Genocchi ${}_{\mathbb{G}}{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ polynomials satisfy the following generating relations:

$$\begin{array}{c}\frac{\mathbb{B}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{\left(q{u}^j\right)}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!},\hfill \\ \frac{\mathbb{E}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{\left(q{u}^j\right)}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{E}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!},\hfill \\ \frac{\mathbb{G}\left(u\right){(1+\lambda )}^{\frac{pu}{\lambda }}}{{\left[\beta -\frac{\left(q{u}^j\right)}{\lambda }log(1+\lambda )\right]}^{\mu }}=\sum _{s=0}^{\infty }{}_{\mathbb{G}}{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\frac{u^s}{s!}.\hfill \end{array}$$

(31)

Next, we establish the determinant representation for these polynomials:

Corollary 3. 

The generalized degenerate 2D Bernoulli polynomials ${}_{\mathbb{G}}{\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ give rise to the determinant of the following form:

$${}_{\mathbb{G}}{\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )=\frac{{(-1)}^s}{{\left({\gamma }_0\right)}^{s+1}}\left|\begin{array}{cccccc}1& {\mathbb{B}}_{1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{B}}_{2,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& \cdots & {\mathbb{B}}_{s-1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{B}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\\ \\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{s-1}& {\gamma }_m\\ \\ 0& {\gamma }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\gamma }_1& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{1}\right){\gamma }_{s-2}& \left(\genfrac{}{}{0pt}{}{s}{1}\right){\gamma }_{s-1}\\ \\ 0& 0& {\gamma }_0& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{2}\right){\gamma }_{s-3}& \left(\genfrac{}{}{0pt}{}{s}{2}\right){\gamma }_{s-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& \left(\genfrac{}{}{0pt}{}{s}{s-1}\right){\gamma }_1\end{array}\right|.$$

Corollary 4. 

The generalized degenerate 2D Euler polynomials ${}_{\mathbb{G}}{\mathbb{E}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ give rise to the determinant of the following form:

$${}_{\mathbb{G}}{\mathbb{E}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )=\frac{{(-1)}^s}{{\left({\gamma }_0\right)}^{s+1}}\left|\begin{array}{cccccc}1& {\mathbb{E}}_{1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{E}}_{2,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& \cdots & {\mathbb{E}}_{s-1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{E}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\\ \\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{s-1}& {\gamma }_m\\ \\ 0& {\gamma }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\gamma }_1& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{1}\right){\gamma }_{s-2}& \left(\genfrac{}{}{0pt}{}{s}{1}\right){\gamma }_{s-1}\\ \\ 0& 0& {\gamma }_0& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{2}\right){\gamma }_{s-3}& \left(\genfrac{}{}{0pt}{}{s}{2}\right){\gamma }_{s-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& \left(\genfrac{}{}{0pt}{}{s}{s-1}\right){\gamma }_1\end{array}\right|.$$

Corollary 5. 

The generalized degenerate 2D Genocchi polynomials ${}_{\mathbb{G}}{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ give rise to the determinant of the following form:

$${}_{\mathbb{G}}{\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )=\frac{{(-1)}^s}{{\left({\gamma }_0\right)}^{s+1}}\left|\begin{array}{cccccc}1& {\mathbb{G}}_{1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{G}}_{2,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& \cdots & {\mathbb{G}}_{s-1,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )& {\mathbb{G}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )\\ \\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{s-1}& {\gamma }_m\\ \\ 0& {\gamma }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\gamma }_1& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{1}\right){\gamma }_{s-2}& \left(\genfrac{}{}{0pt}{}{s}{1}\right){\gamma }_{s-1}\\ \\ 0& 0& {\gamma }_0& \cdots & \left(\genfrac{}{}{0pt}{}{s-1}{2}\right){\gamma }_{s-3}& \left(\genfrac{}{}{0pt}{}{s}{2}\right){\gamma }_{s-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& \left(\genfrac{}{}{0pt}{}{s}{s-1}\right){\gamma }_1\end{array}\right|.$$

5. Conclusions

This study builds on the previous work by [19] on degenerate 2D Appell polynomials; moreover, it introduces a new family of generalized degenerate 2D Appell polynomials with the notation of ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$. In Section 2 of this research, the operational rule in terms of Theorem 1, and the generating expression in terms of Theorem 2, for these polynomials are provided together with an inference of their special cases in the form of remarks. In Section 3, the explicit summation formula for ${}_{\mathbb{G}}{\mathbb{R}}_{s,\mu }^{\left[j\right]}(p,q;\lambda ,\beta )$ is firmly established, and the determinant form for the generalized family is generated. Further, the recurrence relations of the generalized degenerate 2D Appell polynomials are also obtained. In Section 4, the paper demonstrates the broad applicability of the results obtained in Section 2. It presents equivalent results for degenerate 2D Bernoulli, 2D Euler, and 2D Genocchi polynomials. These generalized degenerate special polynomials, which are associated with Appell polynomials, find extensive applications in the fields of mathematics and physics. Notably, they have natural connections to quantum mechanics and probability theory, where they can be linked to the normal distribution—a fundamental concept in probability theory. Furthermore, these polynomials serve as a basis for approximating the functions in approximation theory, thereby making them valuable in numerical analysis. Moreover, in the realm of statistical mechanics, Hermite polynomials—which are a special case of these polynomials—play a crucial role in calculating the partition function and thermodynamic properties of ideal gases. Additionally, they find utility in Fourier analysis for decomposing functions into orthogonal functions.

Operational techniques are vital in the development of novel families of functional equations and for gaining insights into their characteristics; thus, they encompass both regular and generalized special functions. Scholars, including Dattoli and various others, have emphasized the significance of employing operational methods, particularly when addressing the explicit solutions for families of partial differential equations such as the Heat and D’Alembert types, as well as in exploring their wide-ranging applications. For the notable contributions by researchers like Dattoli and others see, for instance [3,30,31]. These studies serve to illustrate the essential role of operational methods in the analysis of special functions and their practical implications.

Future research endeavors can focus on the exploration of symmetric identities and the investigation of the ${\Delta }_h$ properties of the aforementioned polynomials. Additionally, delving into implicit summation equations as potential avenues for obtaining additional data could prove to be beneficial for further analysis. Further, the verification of the monomiality principle and a sequence of upward and downward operators may also be taken as worthy offuture observation.