Two-iterated degenerate Appell polynomials: properties and applications

Abstract

In the development of hybrid special polynomials, it is essential to incorporate the monomiality principle, operational rules, and other related properties. This research study adopts this approach and introduces a novel polynomial type called the two-iterated degenerate Appell polynomials, utilizing the monomiality principle. This study yields new findings that align with previous research efforts. The study presents explicit formulas and key properties of these polynomials, while also establishing connections with other polynomial types such as the Bernoulli, Euler, and Genocchi polynomials. These connections facilitate the derivation of additional results. By employing the monomiality principle and considering the aforementioned polynomials as initial members of the Appell family, this study contributes to the expansion of knowledge in the field of special polynomials.

Keywords:

• MSC (2020): 33E20
• 33B10
• 33E30
• 11T23 determinant form explicit forms
• monomiality principle
• two-iterated degenerate Appell polynomials

1. Introduction and preliminaries

Special functions hold a pivotal significance across a spectrum of mathematical, physical, and engineering domains, as well as in various scientific disciplines. These functions are systematically defined as solutions to distinct categories of mathematical challenges that recurrently emerge in diverse fields of study. Consequently, special functions offer a robust and adaptable set of mathematical tools, indispensable for addressing intricate problems encountered in the realms of science and engineering. The crux of their importance lies in the capability of special functions to articulate solutions to mathematical problems that resist expression through elementary functions. This unique quality positions them as indispensable assets in the modeling and comprehension of an extensive array of both physical and mathematical phenomena. In essence, special functions stand as a cornerstone, facilitating the exploration and resolution of complex issues that permeate the landscape of scientific and engineering disciplines.

There are several important special polynomial sequences in mathematics, each with its unique properties and applications. See, for instance Hermite Polynomials, Legendre Polynomials, Chebyshev Polynomials, Laguerre Polynomials, Jacobi Polynomials, Zernike Polynomials. These polynomial sequences are just a subset of the many special polynomial families that play crucial roles in various branches of mathematics, physics, engineering, and other scientific disciplines. They often arise as solutions to specific differential equations and provide a powerful mathematical framework for understanding and solving diverse problems.

One of the important and the most significant class of polyniomial sequences is the Appell polynomial sequence (Appell, 1880). These sequences have a wide range of applications in theoretical physics, mathematics, approximation theory, and related areas. Additionally, the entire collection of Appell sequences forms an abelian group due to their closure under umbral polynomial sequence composition.

Appell polynomial sequences hold significant importance among polynomial sequences and are found in various mathematical fields, such as engineering sciences, theoretical physics, engineering mathematics, medical sciences and approximation theory (Appell, 1880). Additionally, Appell polynomials fulfill all the axioms of an abelian group when composed together. Further, they have various applications in different fields of mathematics, including combinatorics, algebraic geometry, and number theory. These polynomials have significant applications see for instance, Appell polynomials are used to express solutions to partial differential equations in terms of power series. In particular, the Hermite, Laguerre, and Legendre polynomials are examples of Appell polynomials that have important applications in mathematical physics. Appell (Appell, 1880) in the eighteenth century presented sequences of polynomials ${\mathcal{T}}_n(u),$ which hold the relation: (1.1) $$\frac{d}{\mathit{\text{du}}}{\mathcal{T}}_n(u)=n{\mathcal{T}}_{n-1}(u),n\in {\mathbb{N}}_0,$$(1.1) and possesses the generating relation listed below: (1.2) $$\alpha (\xi )\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left(u\xi \right)=\sum _{n=0}^{\infty }{\mathcal{T}}_n(u)\frac{{\xi }^n}{n!},$$(1.2) where (1.3) $$\alpha (\xi )=\sum _{n=0}^{\infty }{\alpha }_n\frac{{\xi }^n}{n!},{\alpha }_0\ne 0$$(1.3) is a power series expansion of $\alpha (\xi ).$

Special functions, including polynomials, have a wide range of applications in mathematics, physics, and engineering fields, including thermodynamics, information theory, and statistical mechanics. Special polynomials, such as Hermite, Laguerre, and Jacobi polynomials, are used to calculate entropy in different systems. For example, Hermite polynomials are used in the calculation of the quantum mechanical entropy of a system, while Laguerre polynomials are used in the calculation of the entropy of a system with a non-negative definite matrix.

Numerous authors have been actively engaged in the exploration and elucidation of diverse properties associated with degenerate special polynomials. For instance, Ryoo and collaborators have made significant contributions in this realm, as evidenced by their works (Kim, 2019; Kim, Yao, Kim, & Jang, 2018; Kim, Kim, & Lee, 2019; Muhyi, 2022; Ryoo, 2012, 2013; Ryoo, 2016a, 2016b; Yasmin & Muhyi, 2019). More recently, Wani and colleagues have introduced a series of doped polynomials of a specific type, delving into their distinct characteristics and behaviors, particularly pertinent from an engineering standpoint. Their works include (Khan, Nahid, & Riyasat, 2019, 2021; Khan and Wani, 2018, 2021a; Nahid, Alam, & Choi, 2020; Nahid and Ryoo, 2022; Nahid and Ali, 2022; Wani, Khan, & Naikoo, 2020; Wani and Nahid, 2022; Wani and Nisar, 2020).

Key features explored in these investigations encompass determinant forms, operational formalism, approximation properties, poly forms, degenerate forms, summation formulas, approximation quality, explicit and implicit formulae, generating expressions, and more. The examination of degenerate forms of special functions holds paramount importance in comprehending the mathematical underpinnings of various physical phenomena. Notably, these degenerate polynomials play a crucial role in describing the behavior of quantum systems, such as the quantum harmonic oscillator.

The versatile applications of these functions extend across a wide spectrum of scientific and engineering domains, with their significance duly acknowledged in both theoretical mathematics and practical contexts. The ongoing progress in the theory of special functions is anticipated to yield future discoveries and breakthroughs, further enriching our understanding and application of these essential mathematical constructs.

In recent years, Hwang and Ryoo (2020) and Hwang, Seol, and Ryoo (2020) introduced degenerate forms of Hermite polynomials with two and three variables: (1.4) $$\sum _{k=0}^{\infty }{\mathcal{J}}_k(j_1,j_2;\vartheta )\frac{{\xi }^k}{k!}={(1+\vartheta )}^{\xi (\frac{j_1+j_2\xi }{\vartheta })},$$(1.4) and (1.5) $$\sum _{k=0}^{\infty }{\mathcal{F}}_k(j_1,j_2,j_3;\vartheta )\frac{{\xi }^k}{k!}={(1+\vartheta )}^{\xi \left(\frac{j_1+j_2\xi +j_3{\xi }^2}{\vartheta }\right)}.$$(1.5)

 In recent years, Steffenson (Steffensen, 1941) introduced the concept of poweroids in 1941, particularly focusing on the idea of monomiality. Subsequently, Dattoli (2000) and Dattoli, Ricci, Cesarano, and Vázquez (2003) expanded on this notion, transforming it into a crucial tool for investigating exceptional polynomials. Monomiality involves expressing a polynomial set in terms of monomials, the foundational units of polynomials. This approach enhances our comprehension of the polynomial set and its inherent properties.

The operators $\widehat{\mathcal{M}}$ and $\widehat{\mathcal{D}}$ play a pivotal role in the study of special polynomials, serving as multiplicative and derivative operators for the polynomial set $b_k(j_1)$ where $k\in \mathbb{N}.$ The multiplicative property of $\widehat{\mathcal{M}}$ is expressed by the equation (1.6) $$b_{k+1}(j_1)=\widehat{\mathcal{M}}\{b_k(j_1)\},$$(1.6) which generates a new polynomial $b_{k+1}(j_1)$ from the preceding polynomial $b_k(j_1).$ Similarly, the derivative property of $\widehat{\mathcal{D}}$ is conveyed by the equation (1.7) $$k b_{k-1}(j_1)=\widehat{\mathcal{D}}\{b_k(j_1)\},$$(1.7) generating a new polynomial $b_{k-1}(j_1)$ by taking the derivative of $b_k(j_1)$ and multiplying it by the coefficient k.

The incorporation of the monomiality principle and operational rules into the study of special polynomials has led to the development of hybrid special polynomials, possessing unique properties useful in solving various mathematical problems. The exploration of these hybrid special polynomials is an active area of research, finding widespread applications in physics, engineering, and computer science.

These equations and properties are integral components of the theory of quasi-monomials and Weyl groups, with applications spanning various branches of mathematics and physics, including representation theory, algebraic geometry, and quantum field theory. The set of polynomials ${\{b_k(j_1)\}}_{m\in \mathbb{N}},$ in view of operators represented by expressions (1.6)(1.6) $$b_{k+1}(j_1)=\widehat{\mathcal{M}}\{b_k(j_1)\},$$(1.6) and (1.7)(1.7) $$k b_{k-1}(j_1)=\widehat{\mathcal{D}}\{b_k(j_1)\},$$(1.7) , is termed quasi-monomial and satisfies the axioms (1.8) $$[\widehat{\mathcal{D}},\widehat{\mathcal{M}}]=\widehat{\mathcal{D}}\widehat{\mathcal{M}}-\widehat{\mathcal{M}}\widehat{\mathcal{D}}=\widehat{1},$$(1.8) thus forming a Weyl group.

Particularly, $b_k(j_1)$ satisfies the differential equation (1.9) $$\widehat{\mathcal{M}}\widehat{\mathcal{D}}\{b_k(j_1)\}=k b_k(j_1),$$(1.9) implying that the quasi-monomials are eigenfunctions of the operator $\widehat{\mathcal{M}}\widehat{\mathcal{D}}$ with eigenvalue k. This differential equation can be explicitly solved for certain choices of $\widehat{\mathcal{M}}$ and $\widehat{\mathcal{D}},$ leading to explicit expressions for the quasi-monomials, as represented by (1.10) $$b_k(j_1)={\widehat{\mathcal{M}}}^k\left\{1\right\},$$(1.10) with the initial condition $b_0(j_1)=1.$ Equation (1.10)(1.10) $$b_k(j_1)={\widehat{\mathcal{M}}}^k\left\{1\right\},$$(1.10) provides a recursive method for computing the quasi-monomials by repeated application of the operator $\widehat{\mathcal{M}}$ to the identity operator $\widehat{I}.$

Furthermore, expressing $b_k(j_1)$ in terms of an exponential form, we have (1.11) $$e^{\xi \widehat{\mathcal{M}}}\left\{1\right\}=\sum _{k=0}^{\infty }{b}_k(j_1)\frac{{\xi }^k}{k!},\left|\xi \right|<\infty ,$$(1.11) utilizing the identity (1.10)(1.10) $$b_k(j_1)={\widehat{\mathcal{M}}}^k\left\{1\right\},$$(1.10) .

The generating relation (1.11)(1.11) $$e^{\xi \widehat{\mathcal{M}}}\left\{1\right\}=\sum _{k=0}^{\infty }{b}_k(j_1)\frac{{\xi }^k}{k!},\left|\xi \right|<\infty ,$$(1.11) represents the quasi-monomials as a power series, where the coefficients are determined by the quasi-monomials. This expression facilitates precise computations of the quasi-monomials and finds applications in investigating specific special functions and their properties.

Therefore, the theory of quasi-monomials and Weyl groups serves as a potent analytical tool for examining the structure and characteristics of particular function families. Its significance extends to diverse applications in both mathematics and physics.

These findings have motivated us to create two-iterated degenerate Appell polynomials, and we have established their monomiality principle by determining both their explicit and implicit forms. The remainder of the article is organized as follows: Section 2 outlines the construction of these polynomials and investigates some of their fundamental properties, while Section 3 establishes the quasi-monomial characteristics of these polynomials. In Section 4, we examine a few specific members of this polynomial family and discuss their related findings. The article is primarily focused on exploring the properties and potential applications of these newly created polynomials.

2. Two-iterated degenerate Appell polynomials

In this section, we explore a novel category of two-iterated degenerate Appell polynomials and identify various characteristics associated with these polynomials. Through the use of the generating function concept, we provide a definition for the two-iterated degenerate Appell polynomials: (2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1)

For, $\alpha (\xi )=1,$ the two-iterated degenerate Appell polynomials reduce to degenerate Appell polynomials (Riyasat, Nahid, & Khan, 2023) represented by (2.2) $$\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.2)

Also, for $\alpha (\xi )=1=\beta (\xi ),$ the two-iterated degenerate Appell polynomials reduce to the associate degenerate polynomials (Riyasat, Nahid, & Khan, 2023) represented by (2.3) $${(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{R}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.3)

Further, for ${(1+\nu )}^{\frac{u\xi }{\nu }}\to \hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}(u\xi ),$ it is evident that Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) reduces to two-iterated Appell polynomials (Khan & Wani, 2021b): (2.4) $$\alpha (\xi )\beta (\xi )\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}(u\xi )=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u)\frac{{\xi }^n}{n!}.$$(2.4)

Moreover, for $u\to 0,$ it is evident that Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) reduces to two-iterated Appell numbers: (2.5) $$\alpha (\xi )\beta (\xi )=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(0;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}\frac{{\xi }^n}{n!}.$$(2.5)

Theorem 2.1.

The two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu ),n\in \mathbb{N}$ is determined by the power series expansion of the product $\alpha (\xi ),\beta (\xi )$ and ${(1+\nu )}^{\frac{u\xi }{\nu }}={\sum }_{n=0}^{\infty }{\mathcal{T}}_n^{[2]}(u;\nu )\frac{{\xi }^n}{n!}$, that is (2.6) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}={\mathcal{T}}_0^{\left[2\right]}(u;\nu )+{\mathcal{T}}_1^{\left[2\right]}(u;\nu )\frac{\xi }{1!}+{\mathcal{T}}_2^{\left[2\right]}(u;\nu )\frac{{\xi }^2}{2!}+\cdots +{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}+\cdots .$$(2.6)

Proof.

By utilizing the Newton series for finite differences and expanding $\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}$ at u = 0, we can order the resulting product of functions w.r.t. the powers of ξ. This process leads us to discover the special polynomials ${\mathcal{T}}_n^{[2]}(u;\nu ),$ which are expressed as the coefficients of $\frac{{\xi }^n}{n!}$ in Equation (2.6)(2.6) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}={\mathcal{T}}_0^{\left[2\right]}(u;\nu )+{\mathcal{T}}_1^{\left[2\right]}(u;\nu )\frac{\xi }{1!}+{\mathcal{T}}_2^{\left[2\right]}(u;\nu )\frac{{\xi }^2}{2!}+\cdots +{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}+\cdots .$$(2.6) . These polynomials are the generating function of two-iterated degenerate Appell polynomials.□

We will now establish the explicit forms of the two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ by proving the following theorem:

Theorem 2.2.

The two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ satisfy the following explicit form: (2.7) $${\mathcal{T}}_n^{\left[2\right]}(u;\nu )=\sum _{l=0}^{\left[n\right]}\left(\begin{array}{c}n\\ l\end{array}\right){\mathcal{T}}_{n-l}^{\left[2\right]}{\left(\frac{\mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l{u}^l.$$(2.7)

Proof.

Inserting expression Equation (2.5)(2.5) $$\alpha (\xi )\beta (\xi )=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(0;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}\frac{{\xi }^n}{n!}.$$(2.5) and expression: (2.8) $${(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{l=0}^{\infty }{\left(\frac{u \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l \frac{{\xi }^l}{l!},$$(2.8) in generating relation Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) , it follows that (2.9) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}\frac{{\xi }^n}{n!} \sum _{l=0}^{\infty }{\left(\frac{u \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l \frac{{\xi }^l}{l!}.$$(2.9)

Further simplifying the above expression, we find (2.10) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}{\left(\frac{u \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l\hspace{0.5em}\frac{{\xi }^{n+l}}{n!l!}.$$(2.10)

Using (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) in the l.h.s. of the Equation (2.10)(2.10) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}{\left(\frac{u \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l\hspace{0.5em}\frac{{\xi }^{n+l}}{n!l!}.$$(2.10) , it follows that $$\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}{\left(\frac{u \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l \frac{{\xi }^{n+l}}{n!l!}.$$

Demonstrating the validity of statement Equation (2.5)(2.5) $$\alpha (\xi )\beta (\xi )=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(0;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}\frac{{\xi }^n}{n!}.$$(2.5) involves substituting n with n—l in the right-hand side of the equation using the Cauchy product rule. By equating the coefficients of corresponding powers of ξ on both sides of the resultant equation, we can establish the truth of assertion Equation (2.5)(2.5) $$\alpha (\xi )\beta (\xi )=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(0;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}\frac{{\xi }^n}{n!}.$$(2.5) .□

Moreover, we ascertain explicit expressions for the two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ by validating the outcomes presented below:

Theorem 2.3.

The two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ satisfy the following explicit form: (2.11) $${\mathcal{T}}_n^{\left[2\right]}(u;\nu )=\sum _{k=0}^{\left[n\right]}\left(\begin{array}{c}n\\ k\end{array}\right){\alpha }_k {\mathcal{T}}_{n-k}(u;\nu ).$$(2.11)

Proof.

Inserting expressions (1.3)(1.3) $$\alpha (\xi )=\sum _{n=0}^{\infty }{\alpha }_n\frac{{\xi }^n}{n!},{\alpha }_0\ne 0$$(1.3) and (2.2)(2.2) $$\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.2) in generating relation Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) , it follows that (2.12) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{k=0}^{\infty }{\alpha }_k\frac{{\xi }^k}{k!} \sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.12)

Next, using the r.h.s. of expression (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) in above expression (2.12)(2.12) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{k=0}^{\infty }{\alpha }_k\frac{{\xi }^k}{k!} \sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.12) , we have (2.13) $$\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}=\sum _{k=0}^{\infty }{\alpha }_k\frac{{\xi }^k}{k!} \sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.13)

Replacing n by n—k and using the C.P. (Cauchy Product) rule in the r.h.s. of previous equation, it follows that (2.14) $$\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }\sum _{k=0}^{\left[n\right]}\left(\begin{array}{c}n\\ k\end{array}\right){\alpha }_k{\mathcal{T}}_{n-k}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.14)

Thus, equating the coefficients of the same powers of ξ on both sides, assertion Equation (2.11)(2.11) $${\mathcal{T}}_n^{\left[2\right]}(u;\nu )=\sum _{k=0}^{\left[n\right]}\left(\begin{array}{c}n\\ k\end{array}\right){\alpha }_k {\mathcal{T}}_{n-k}(u;\nu ).$$(2.11) is proved. □

In the forthcoming section, we will explore the quasi-monomial characteristics inherent in the two-iterated degenerate Appell polynomials. Our focus will be on investigating the behavior of these polynomials when subjected to multiplication by a monomial function expressed as ξ^k. This examination is intended to provide enhanced insights into the inherent properties and behavior of these polynomials.

3. Quasi-monomial properties

In this section, we aim to validate the quasi-monomial properties of the two-iterated degenerate Appell polynomials by demonstrating the following outcomes:

Theorem 3.1.

The two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ exhibit the following properties with respect to multiplicative and derivative operators: (3.1) $${\mathcal{T}}_{n+1}^{\left[2\right]}(u,\nu )=\widehat{{\mathcal{M}}_{{\mathcal{T}}^{\left[2\right]}}}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=(\frac{u}{\eta }+\frac{\alpha \prime \left(\eta \frac{\partial }{\partial u}\right)}{\alpha \left(\eta \frac{\partial }{\partial u}\right)}+\frac{\beta \prime \left(\eta \frac{\partial }{\partial u}\right)}{\beta \left(\eta \frac{\partial }{\partial u}\right)})\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\},$$(3.1) and (3.2) $$n {\mathcal{T}}_{n-1}^{\left[2\right]}(u;\nu )=\widehat{{\mathcal{D}}_{{\mathcal{T}}^{\left[2\right]}}}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=\eta \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}.$$(3.2)

Proof.

Analyzing the two-iterated degenerate Appell polynomials by taking the derivatives of their generating expression (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) with respect to ξ, this yields (3.3) $$\frac{\partial }{\partial \xi }\{\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}\}=\left(\frac{\alpha \prime (\xi )}{\alpha (\xi )}+\frac{\beta \prime (\xi )}{\beta (\xi )}+\frac{\mathrm{\text{log}}(1+\lambda )}{\lambda }u\right)\{\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}\}=n {\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^{n-1}}{n!}.$$(3.3)

In consideration of identity: $$\frac{\partial }{\partial u}\{\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}\}=\left(\frac{\mathrm{\text{log}}(1+\lambda )}{\lambda } \xi \right)\{\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}\},$$ which in view of expression (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) becomes (3.4) $$\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )} \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}=\xi \{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}.$$(3.4)

Using $\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )}=\eta$ in (3.4)(3.4) $$\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )} \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}=\xi \{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}.$$(3.4) and using Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) in Equation (3.3)(3.3) $$\frac{\partial }{\partial \xi }\{\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}\}=\left(\frac{\alpha \prime (\xi )}{\alpha (\xi )}+\frac{\beta \prime (\xi )}{\beta (\xi )}+\frac{\mathrm{\text{log}}(1+\lambda )}{\lambda }u\right)\{\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}\}=n {\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^{n-1}}{n!}.$$(3.3) with n replaced by n + 1 in the last summation, it follows that (3.5) $$\begin{array}{ll}\left(\frac{\alpha \prime (\eta \frac{\partial }{\partial u})}{\alpha (\eta \frac{\partial }{\partial u})}+\frac{\beta \prime (\eta \frac{\partial }{\partial u})}{\beta (\eta \frac{\partial }{\partial u})}+\frac{\mathrm{\text{log}}(1+\lambda )}{\lambda }u\right)\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}={\mathcal{T}}_{n+1}^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.& \end{array}$$(3.5)

By comparing the coefficients of similar powers of $\frac{{\xi }^n}{n!}$ in the equation obtained from taking the derivatives of the generating expression with respect to ξ and using Equation (1.6)(1.6) $$b_{k+1}(j_1)=\widehat{\mathcal{M}}\{b_k(j_1)\},$$(1.6) , we can establish the validity of assertion Equation (3.1)(3.1) $${\mathcal{T}}_{n+1}^{\left[2\right]}(u,\nu )=\widehat{{\mathcal{M}}_{{\mathcal{T}}^{\left[2\right]}}}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=(\frac{u}{\eta }+\frac{\alpha \prime \left(\eta \frac{\partial }{\partial u}\right)}{\alpha \left(\eta \frac{\partial }{\partial u}\right)}+\frac{\beta \prime \left(\eta \frac{\partial }{\partial u}\right)}{\beta \left(\eta \frac{\partial }{\partial u}\right)})\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\},$$(3.1) .

Also, in view of identity Equation (3.4)(3.4) $$\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )} \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}=\xi \{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}.$$(3.4) , we find (3.6) $$\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )} \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}=\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^{n+1}}{n!}\}.$$(3.6)

By substituting n with n − 1 in the right-hand side of the equation obtained previously, we can obtain the following result: $$\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )} \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}=n \{{\mathcal{T}}_{n-1}^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\}.$$

By comparing the coefficients of the same powers of ξ on both sides of the equation obtained above, we can establish the validity of assertion Equation (3.2)(3.2) $$n {\mathcal{T}}_{n-1}^{\left[2\right]}(u;\nu )=\widehat{{\mathcal{D}}_{{\mathcal{T}}^{\left[2\right]}}}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=\eta \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}.$$(3.2) .□

In order to find the differential equation satisfied by two-iterated degenerate Appell polynomials ${\mathcal{T}}^{[2]}n(u;\nu ),$ the succeeding result is demonstrated:

Theorem 3.2.

The differential equation satisfied by two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ is given by: (3.7) $$\begin{array}{l}\left(\right(u+\eta \frac{\alpha \prime \left(\eta \frac{\partial }{\partial u}\right)}{\alpha \left(\eta \frac{\partial }{\partial u}\right)}+\eta \frac{\beta \prime \left(\eta \frac{\partial }{\partial u}\right)}{\beta \left(\eta \frac{\partial }{\partial u}\right)})\frac{\partial }{\partial u}-n)\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=0.\end{array}$$(3.7)

Proof.

Inserting Equations (3.1)(3.1) $${\mathcal{T}}_{n+1}^{\left[2\right]}(u,\nu )=\widehat{{\mathcal{M}}_{{\mathcal{T}}^{\left[2\right]}}}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=(\frac{u}{\eta }+\frac{\alpha \prime \left(\eta \frac{\partial }{\partial u}\right)}{\alpha \left(\eta \frac{\partial }{\partial u}\right)}+\frac{\beta \prime \left(\eta \frac{\partial }{\partial u}\right)}{\beta \left(\eta \frac{\partial }{\partial u}\right)})\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\},$$(3.1) and (3.2)(3.2) $$n {\mathcal{T}}_{n-1}^{\left[2\right]}(u;\nu )=\widehat{{\mathcal{D}}_{{\mathcal{T}}^{\left[2\right]}}}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=\eta \frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}.$$(3.2) in Equation (1.9)(1.9) $$\widehat{\mathcal{M}}\widehat{\mathcal{D}}\{b_k(j_1)\}=k b_k(j_1),$$(1.9) , it follows that $$(\frac{u}{\eta }+\frac{\alpha \prime \left(\eta \frac{\partial }{\partial u}\right)}{\alpha \left(\eta \frac{\partial }{\partial u}\right)}+\frac{\beta \prime \left(\eta \frac{\partial }{\partial u}\right)}{\beta \left(\eta \frac{\partial }{\partial u}\right)})\eta \left(\frac{\partial }{\partial u}\right)\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=n \{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}.$$

After simplification, the differential equation for the two-iterated degenerate Appell polynomials is proven, thus confirming Equation (3.7)(3.7) $$\begin{array}{l}\left(\right(u+\eta \frac{\alpha \prime \left(\eta \frac{\partial }{\partial u}\right)}{\alpha \left(\eta \frac{\partial }{\partial u}\right)}+\eta \frac{\beta \prime \left(\eta \frac{\partial }{\partial u}\right)}{\beta \left(\eta \frac{\partial }{\partial u}\right)})\frac{\partial }{\partial u}-n)\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}=0.\end{array}$$(3.7) .

Summation formulae, also known as summation identities or series identities, play a crucial role in mathematics and its applications. These formulas provide a concise and efficient way to represent the sum of a sequence of numbers, and they find applications in various branches of mathematics, physics, engineering, computer science, and other scientific disciplines. These formulae are fundamental tools in mathematical analysis for manipulating and simplifying series. They allow mathematicians to express complex series in a more compact and manageable form, aiding in the study of convergence, divergence, and other properties of series. Further, in computer science and algorithm analysis, summation formulae are used to analyze the time complexity of algorithms. They help express the running time of algorithms in terms of summations, providing insights into their efficiency and performance. Also, these formulae are often employed in probability and statistics to represent the sums of probabilities or expected values. In the context of probability distributions, these formulas help analyze the behavior of random variables and are significant in number theory, where they are used to express and manipulate various arithmetic functions. Examples include the formulas for the sum of the first n natural numbers, the sum of squares, and the sum of cubes.

Thus, the summation formulae provide a powerful and versatile tool for expressing and manipulating a series of numbers. Their significance extends across various mathematical and scientific disciplines, contributing to the understanding, analysis, and solution of problems in diverse fields.

Here, we derive the summation formulae of the two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ by providing the succeeding results:

Theorem 3.3.

For, the two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$, the succeeding summation formulae holds true: (3.8) $${\mathcal{T}}_n^{\left[2\right]}(u+w;\nu )=\left(\begin{array}{l}n\\ k\end{array}\right){\mathcal{T}}_{n-k}^{\left[2\right]}(u;\nu ){\mathcal{R}}_k(w;\nu ).$$(3.8)

Proof.

Replacing $u\to u+w$ in the expression (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) , we have (3.9) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{(u+w)\xi }{\nu }}=\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}{(1+\nu )}^{\frac{w\xi }{\nu }}={\mathcal{T}}_n^{\left[2\right]}(u+w;\nu )\frac{{\xi }^n}{n!}.$$(3.9)

Using expressions (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) and (2.3)(2.3) $${(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{R}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.3) in previous expression, it follows that (3.10) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}{(1+\nu )}^{\frac{w\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}\sum _{k=0}^{\infty }{\mathcal{R}}_k(w;\nu )\frac{{\xi }^k}{k!}.$$(3.10)

Using the Cauch product rule in the r.h.s. of the above expression, assertion Equation (3.8)(3.8) $${\mathcal{T}}_n^{\left[2\right]}(u+w;\nu )=\left(\begin{array}{l}n\\ k\end{array}\right){\mathcal{T}}_{n-k}^{\left[2\right]}(u;\nu ){\mathcal{R}}_k(w;\nu ).$$(3.8) is obtained.□

Theorem 3.4.

For, the two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$, the succeeding summation formulae holds true: (3.11) $${\mathcal{T}}_n^{\left[2\right]}(u;\nu )=\sum _{k=0}^n\left(\begin{array}{l}n\\ k\end{array}\right){\alpha }_{k,h} {\mathcal{T}}_{n-k}(u;\nu ).$$(3.11)

Proof.

Using expression (2.2)(2.2) $$\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(2.2) and the expression (3.12) $$\alpha (\xi )={\alpha }_{k,\nu } \frac{{\xi }^k}{k!}$$(3.12) in the l.h.s. of expression (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) , we have (3.13) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{k=0}^{\infty }{\alpha }_{k,\nu } \frac{{\xi }^k}{k!}\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(3.13)

Using the r.h.s. of the expression (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) in previous expression, it follows that (3.14) $$\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}=\sum _{k=0}^{\infty }{\alpha }_{k,\nu } \frac{{\xi }^k}{k!}\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}.$$(3.14)

Using the Cauch product rule in the r.h.s. of the above expression, assertion Equation (3.11)(3.11) $${\mathcal{T}}_n^{\left[2\right]}(u;\nu )=\sum _{k=0}^n\left(\begin{array}{l}n\\ k\end{array}\right){\alpha }_{k,h} {\mathcal{T}}_{n-k}(u;\nu ).$$(3.11) is obtained. □

Theorem 3.5.

The two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ satisfy the following summation formulae: (3.15) $${\mathcal{T}}_n^{\left[2\right]}(u+w;\nu )=\sum _{l=0}^{\left[n\right]}\left(\begin{array}{l}n\\ l\end{array}\right){\mathcal{T}}_{n-l}^{\left[2\right]}(u;\nu ){\left(\frac{\mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l{w}^l.$$(3.15)

Proof.

Replacing $u\to u+w$ Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) and using the expression: (3.16) $${(1+\nu )}^{\frac{w\xi }{\nu }}=\sum _{l=0}^{\infty }(\frac{w \mathrm{\text{log}}(1+\nu )}{\nu }{)}^l \frac{{\xi }^l}{l!},$$(3.16) in generating relation Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) , it follows that (3.17) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{(u+w)\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!} \sum _{l=0}^{\infty }(\frac{w \mathrm{\text{log}}(1+\nu )}{\nu }{)}^l \frac{{\xi }^l}{l!}.$$(3.17)

Further simplifying above expression, we find (3.18) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{(u+w)\xi }{\nu }}=\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu ){\left(\frac{w \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l \frac{{\xi }^{n+l}}{n!l!}.$$(3.18)

Using (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) in the l.h.s. of the Equation (3.18)(3.18) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{(u+w)\xi }{\nu }}=\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu ){\left(\frac{w \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l \frac{{\xi }^{n+l}}{n!l!}.$$(3.18) , it follows that $$\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u=w;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu ){\left(\frac{w \mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l \frac{{\xi }^{n+l}}{n!l!}.$$

Demonstrating the validity of statement Equation (3.15)(3.15) $${\mathcal{T}}_n^{\left[2\right]}(u+w;\nu )=\sum _{l=0}^{\left[n\right]}\left(\begin{array}{l}n\\ l\end{array}\right){\mathcal{T}}_{n-l}^{\left[2\right]}(u;\nu ){\left(\frac{\mathrm{\text{log}}(1+\nu )}{\nu }\right)}^l{w}^l.$$(3.15) involves substituting n with n—l in the right-hand side of the equation using the Cauchy product rule. By equating the coefficients of corresponding powers of ξ on both sides of the resultant equation.□

4. Operational formalism and determinant form

Operational techniques and determinant forms, play a significant role in the study of entropy in various fields, including statistical mechanics, information theory, and quantum mechanics. The use of operational and determinant techniques has led to the development of a powerful tool called the density matrix method, which allows one to calculate the entropy of a system in terms of the eigenvalues of the density matrix. In information theory, operational techniques have been used to derive the Shannon entropy formula, which measures the uncertainty of a random variable. In quantum mechanics, operational techniques have been used to study the entropy of entanglement, which characterizes the degree of correlation between the states of two quantum systems. Operational methodologies remain prevalent in contemporary applications within mathematical physics, quantum mechanics, and classical optics. These methods continue to serve as robust and efficient tools for research, as exemplified in references such as (Bretti, Cesarano, & Ricci, 2004; Carlitz, 1959; Dattoli, Ricci, Cesarano, & Vázquez, 2003; Steffensen, 1941). Operational approaches denote mathematical techniques focused on manipulating functions or operators. These approaches are widely used in various fields of mathematical physics, quantum mechanics, and classical optics. As a result, they provide powerful and effective tools for research in these areas. These studies have shown the significance and usefulness of operational approaches in solving complex mathematical problems in various areas of research.

The two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu ),$ which can be expressed using the generating relation Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) are solutions’ of the equation: (4.1) $$\begin{array}{ll}\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )}\frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}& =n{\mathcal{T}}_{n-1}^{\left[2\right]}(u;\nu ),\\ {\mathcal{T}}_n^{\left[2\right]}(u;0)& ={\mathcal{T}}_n^{\left[2\right]}(u),\end{array}$$(4.1) respectively.

 Thus, in view of Equations (4.1)(4.1) $$\begin{array}{ll}\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )}\frac{\partial }{\partial u}\{{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\}& =n{\mathcal{T}}_{n-1}^{\left[2\right]}(u;\nu ),\\ {\mathcal{T}}_n^{\left[2\right]}(u;0)& ={\mathcal{T}}_n^{\left[2\right]}(u),\end{array}$$(4.1) and (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) , it follows that the two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ can be expressed as: (4.2) $$\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}(\frac{\lambda }{\mathrm{\text{log}}(1+\lambda )} \frac{\partial }{\partial u}){\mathcal{T}}_m(u)={\mathcal{T}}_n^{\left[2\right]}(u;\nu ).$$(4.2)

Next, We establish the determinant form of the two-iterated degenerate Appell polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ by using Equation (2.1)(2.1) $$\alpha (\xi )\beta (\xi ){(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(2.1) : (4.3) $$\alpha (\xi )\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!},$$(4.3) where (4.4) $$\alpha (\xi )=\sum _{n=0}^{\infty }{\alpha }_n\frac{{\xi }^n}{n!},{\alpha }_0\ne 0.$$(4.4)

 Thus, using the above expressions, we can write the determinant form satisfied by the polynomials ${\mathcal{T}}_n^{[2]}(u;\nu )$ as:

Theorem 4.1.

The two-iterated degenerate Appell polynomials can be expressed in the form of a determinant as follows: (4.5) $${\mathcal{T}}_n^{\left[2\right]}(u;\nu )=\frac{{(-1)}^n}{{({\gamma }_0)}^{n+1}} \left|\begin{array}{cccccc}1& {\mathcal{T}}_1(u;\nu )& {\mathcal{T}}_2(u;\nu )& \cdots & {\mathcal{T}}_{n-1}(u;\nu )& {\mathcal{T}}_n(u;\nu )\\ & & & & & \\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{n-1}& {\gamma }_n\\ & & & & & \\ 0& {\gamma }_0& \left(\begin{array}{l}2\\ 1\end{array}\right){\gamma }_1& \cdots & \left(\begin{array}{l}n-1\\ 1\end{array}\right){\gamma }_{n-2}& \left(\begin{array}{l}m\\ 1\end{array}\right){\gamma }_{n-1}\\ & & & & & \\ 0& 0& {\gamma }_0& \cdots & \left(\begin{array}{l}n-1\\ 2\end{array}\right){\gamma }_{n-3}& \left(\begin{array}{l}m\\ 2\end{array}\right){\gamma }_{n-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& \left(\begin{array}{l}n\\ n-1\end{array}\right){\gamma }_1\end{array}\right|,$$(4.5)

where ${\gamma }_n,n=0,1,\dots \mathit{\text{are the coefficients}}o\mathrm{f\hspace{0.5em}}\mathit{\text{Maclaurin}}\prime s\mathit{\text{series}} \frac{1}{\alpha (\xi )}.$

Proof.

Taking the product in the r.h.s. while multiplying by $\frac{1}{\alpha (\xi )}={\sum }_{k=0}^{\infty }{\gamma }_k\frac{{\xi }^k}{k!}$ on both sides of the Equation (4.3)(4.3) $$\alpha (\xi )\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )\frac{{\xi }^n}{n!}=\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!},$$(4.3) , we find (4.6) $$\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )=\sum _{k=0}^{\infty }{\gamma }_k\frac{{\xi }^k}{k!}\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu ).$$(4.6)

Thus on usage of the Cauchy product rule in the expression (4.6)(4.6) $$\sum _{n=0}^{\infty }{\mathcal{T}}_n(u;\nu )=\sum _{k=0}^{\infty }{\gamma }_k\frac{{\xi }^k}{k!}\sum _{n=0}^{\infty }{\mathcal{T}}_n^{\left[2\right]}(u;\nu ).$$(4.6) , it gives (4.7) $${\mathcal{T}}_n(u,;\nu )=\sum _{k=0}^m\left(\begin{array}{l}n\\ k\end{array}\right) {\gamma }_k {\mathcal{T}}_{n-K}^{\left[2\right]}(u;\nu ).$$(4.7)

The system comprising n equations involving unknowns ${\mathcal{T}}_n^{[2]}(u;\nu ),n=0,1,2,\dots$ is generated as an outcome of this equality. To solve this system, Cramer’s rule is applied. Exploiting the fact that the denominator corresponds to the determinant of a lower triangular matrix with a determinant of ${({\gamma }_0)}^{n+1}$ and leveraging the transposition of the numerator, the desired result is obtained by substituting the $(i+1)$-th position for the i-th row, where i ranges from 1 to n − 1. □

5. Examples

The Appell polynomial family is a class of special functions that are solutions to certain types of differential equations. The family is parameterized by a function, denoted as $\alpha (\xi ),$ and different members of this family can be obtained by making specific choices for $\alpha (\xi ).$ The significance of this flexibility lies in the ability to tailor the properties and behavior of the Appell polynomials to suit the requirements of various mathematical and scientific problems. The presence of the function $\alpha (\xi )$ in the Appell polynomial family allows for a broad range of possible functions. This parameterization enables mathematicians and scientists to choose or design specific functions that align with the characteristics or constraints of the problem under consideration.

As different problems in mathematics, physics, engineering, and other disciplines may have unique requirements. By selecting an appropriate function for $\alpha (\xi ),$ it is possible to tailor the properties of the resulting Appell polynomial to address the specific needs of a given application. This adaptability is crucial for the versatility of the Appell polynomial family.

The reference to Table 1 underscores the organization and presentation of various members of the Appell polynomial family. The table serves as a comprehensive resource, listing the names of different Appell polynomials, their generating functions, series definitions, and associated numerical values. This organized format facilitates quick access to key information about each member. The generating functions and series definitions provided in Table 1 are essential for understanding the mathematical properties and representations of Appell polynomials. Generating functions offer a concise way to express the polynomial as a power series, while series definitions provide a formal representation. These aspects are crucial for analytical manipulation and solution of mathematical problems. Further, the inclusion of associated numerical values in Table 1 enhances the practical utility of the information. Numerical values provide concrete examples, allowing users to evaluate and apply the Appell polynomials in specific calculations or simulations. This is particularly valuable for researchers and practitioners in various fields.

Table 1. Certain number of the Appell family members.

S. No.	Name of the polynomials and related numbers	$\alpha (\xi )$	Generating expression	Series representation
I.	Bernoulli	$\left(\frac{\xi }{e^{\xi }-1}\right)$	$\left(\frac{\xi }{e^{\xi }-1}\right)e^{u\xi }={\sum }_{k=0}^{\infty }{\mathfrak{B}}_k(u)\frac{{\xi }^k}{k!}$	${\mathfrak{B}}_k(u)={\sum }_{m=0}^k\left(\begin{array}{c}k\\ m\end{array}\right){\mathfrak{B}}_m{\xi }^{k-m}$
	polynomials		$\left(\frac{\xi }{e^{\xi }-1}\right)={\sum }_{k=0}^{\infty }{\mathfrak{B}}_k\frac{{\xi }^k}{k!}$
	and numbers (Arthur, Magnus, Oberhettinger, & Tricomi, 1955)		${\mathfrak{B}}_k:={\mathfrak{B}}_k(0)$
II.	Euler	$\left(\frac{2}{e^{\xi }+1}\right)$	$\left(\frac{2}{e^{\xi }+1}\right)e^{u\xi }={\sum }_{k=0}^{\infty }{\mathfrak{E}}_k(u)\frac{{\xi }^k}{k!}$	${\mathfrak{E}}_k(u)={\sum }_{m=0}^k\left(\begin{array}{c}k\\ m\end{array}\right)\frac{{\mathfrak{E}}_m}{2^m}{\left(\xi -\frac{1}{2}\right)}^{k-m}$
	polynomials		$\frac{2\xi }{e^{2\xi }+1}={\sum }_{k=0}^{\infty }{\mathfrak{E}}_k\frac{{\xi }^k}{k!}$
	and numbers (Arthur, Magnus, Oberhettinger, & Tricomi, 1955)		${\mathfrak{E}}_k:=2^k{\mathfrak{E}}_k\left(\frac{1}{2}\right)$
III.	Genocchi	$\left(\frac{2\xi }{e^{\xi }+1}\right)$	$\left(\frac{2\xi }{e^{\xi }+1}\right)e^{u\xi }={\sum }_{k=0}^{\infty }{\mathfrak{G}}_k(u)\frac{{\xi }^k}{k!}$	${\mathfrak{G}}_k(u)={\sum }_{m=0}^k\left(\begin{array}{c}k\\ m\end{array}\right){\mathfrak{G}}_m{\xi }^{k-m}$
	polynomials		$\frac{2\xi }{e^{\xi }+1}={\sum }_{k=0}^{\infty }{\mathfrak{G}}_k\frac{{\xi }^k}{k!}$
	and numbers (Sandor & Crstici, 2004)		${\mathfrak{G}}_k:={\mathfrak{G}}_k(0)$

The flexibility of the Appell polynomial family and the information provided in Table 1 make these polynomials applicable in diverse fields. They find applications in quantum mechanics, mathematical physics, signal processing, probability theory, and other areas where specialized functions are required to model and solve complex problems. The ability to choose different functions for $\alpha (\xi )$ in the Appell polynomial family, along with the comprehensive information presented in Table 1, makes these polynomials a versatile and powerful tool in the mathematician’s and scientist’s toolkit. They offer a customizable approach to solving problems across various disciplines, providing tailored solutions to specific mathematical challenges.

The Bernoulli, Euler, and Genocchi numbers hold significant importance across diverse mathematical domains, finding myriad applications. Within the realm of the Bernoulli numbers, a sequence of rational numbers, their presence is evident in various mathematical formulas, including the Euler-Maclaurin formula and Bernoulli polynomials. These numbers are instrumental in number theory, combinatorics, and numerical analysis, establishing connections with algebraic geometry and representation theory.

Similarly, the Euler numbers, comprising an integer sequence, emerge in numerous mathematical areas such as algebraic topology, the geometry of smooth manifolds, combinatorics, number theory, and elliptic curve theory. Notably, they contribute significantly to the theory of modular forms, with implications for cryptography and coding theory.

The Genocchi numbers, consisting of integers, feature prominently in diverse combinatorial challenges like counting up-down sequences and labeled rooted trees. Their connections to the Riemann zeta function extend their influence to graph theory and automata theory.

The trigonometric and hyperbolic secant functions are closely intertwined with Euler numbers. The Taylor series expansions of these functions involve Euler numbers and their derivatives, finding applications in various mathematical and physical domains, including signal processing and quantum field theory.

Considering Bernoulli, Euler, and Genocchi polynomials as integral members of the Appell family allows the derivation of diverse members of the two-iterated degenerate Appell polynomials. The Appell family, comprising special functions encapsulating well-known polynomial sequences, extends its scope by initiating with these three polynomials, giving rise to an array of additional polynomials.

The two-iterated degenerate Appell polynomials represent a distinctive subclass within this expanded polynomial family. Through iterative operations applied to the initial Appell family members, novel polynomials exhibiting degenerate behavior are obtained. These polynomials possess unique properties and characteristics, offering fertile ground for exploration through in-depth investigations.

Analyzing the generating expressions of these polynomials, delving into their properties, and establishing connections with other mathematical concepts can yield valuable insights into their behavior and applications across diverse fields. A thorough examination of their generating expressions and an exploration of their properties contribute to a more profound understanding of these polynomial sequences. Thus, different members of ${\mathcal{T}}_n^{[2]}(u;\nu )$ are as two-iterated degenerate Bernoulli polynomials ${\mathcal{B}}_n^{[2]}(u;\nu ),$ two-iterated degenerate Euler polynomials ${\mathcal{E}}_n^{[2]}(u;\nu )$ and two-iterated degenerate Genocchi polynomials ${\mathcal{G}}_n^{[2]}(u;\nu ).$ These polynomials can be generated using the following expressions: (5.1) $${\left(\frac{\xi }{(e^{\xi }-1)}\right)}^2{(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{B}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!},$$(5.1) (5.2) $${\left(\frac{2}{(e^{\xi }+1)}\right)}^2{(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{E}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!},$$(5.2) and (5.3) $${\left(\frac{2\xi }{(e^{\xi }+1)}\right)}^2{(1+\nu )}^{\frac{u\xi }{\nu }}=\sum _{n=0}^{\infty }{\mathcal{G}}_n^{\left[2\right]}(u;\nu )\frac{{\xi }^n}{n!}.$$(5.3)

Similarly, analogous results pertaining to these polynomials can be established using a comparable approach. One such result is the monomiality principle, which delves into the behavior of the polynomials in relation to their monomial coefficients. A thorough examination of the monomiality principle provides researchers with valuable insights into the inherent structure and properties of these polynomials.

Moreover, explicit expressions for the polynomials can be deduced, elucidating a clear representation of their coefficients and terms. This facilitates a more straightforward comprehension and analysis of the polynomials.

Another avenue for exploration involves investigating the differential equations satisfied by these polynomials. The study of the associated differential equations unveils connections between these polynomials and other mathematical concepts. It also allows researchers to scrutinize their behavior under diverse conditions, providing a comprehensive understanding.

Additionally, delving into determinant forms for these polynomials presents an intriguing and valuable research direction. Determinants, mathematical entities describing specific properties of matrices, can be employed to establish connections between these polynomials and linear algebra. This pursuit opens avenues for additional insights and applications in the broader mathematical context.

6. Conclusion

The article discusses both the operational approaches used in mathematical physics and the properties and applications of the two-iterated degenerate Appell polynomials. The use of operational approaches in mathematical physics is explained, including their ability to simplify complex systems for analysis. Then, the specific characteristics of the two-iterated degenerate Appell polynomials are discussed, including their generating relation, explicit forms, quasi-monomial properties, operational rule, and determinant form. The passage concludes by noting that these polynomials have many uses in mathematics and science, particularly in the field of quantum mechanics.

In the future, researchers can conduct further investigations to uncover additional properties and characteristics of the mentioned polynomials. This can involve exploring extended and generalized forms, as well as integral representations. By analyzing these aspects, researchers can gain a deeper understanding of the polynomials and their behavior. Additionally, determinant forms and summation formulas associated with these polynomials present challenges that require new observations and analysis. These mathematical structures may hold valuable insights, and studying them can lead to the development of new techniques and approaches. Therefore, future investigations hold the potential to expand our knowledge of these polynomials, uncover their underlying properties, and address the challenges posed by determinant forms and summation formulas.

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