Degenerate 2D bivariate Appell polynomials: properties and applications

Abstract

The development of certain aspects of hybrid special polynomials after incorporating monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, thus in line with prior facts, our aim is to introduce the degenerate 2D bivariate Appell polynomials ${\mathbb{R}}_j^m(u,v;h)$. Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials. Also, generating relations corresponding to the degenerate 2D bivariate Bernoulli, Euler, and Genocchi polynomials are established, they will further be helpful in establishing the corresponding results for these polynomials.

Keywords:

• Degenerate 2D bivariate Appell polynomials
• monomiality principle
• explicit forms
• determinant form

Mathematics Subject Classifications:

• 33E20
• 33B10
• 33E30
• 11T23

1. Introduction and preliminaries

Polynomial sequences are of interest in enumerative combinatorics, algebraic combinatorics, and applied mathematics. The Laguerre polynomials, Chebyshev polynomials, Legendre polynomials, and Jacobi polynomials are a few polynomial sequences that appear as solutions to particular ordinary differential equations in physics and approximation theory. The most significant polynomial sequences is a class of Appell polynomial sequences [1]. Many applications of the Appell polynomial sequence may be found in theoretical physics, approximation theory, mathematics, and related fields of mathematics. The set of all Appell sequences is closed as a result of umbral polynomial sequence composition. This process turns the collection of all Appell sequences into an abelian group.

Appell [1] in eighteenth century presented sequences of polynomials $R_m\left(u\right)$, which hold the relation: (1) $\begin{array}{r}\frac{\text{d}}{\text{d}u}{R}_m\left(u\right)=m{R}_{m-1}\left(u\right),m\in {\mathbb{N}}_0\end{array}$(1) and possesses the generating relation listed below: (2) $\begin{array}{r}R\left(t\right)\exp \left(ut\right)=\sum _{k=0}^{\mathrm{\infty }}{R}_k\left(u\right)\frac{t^k}{k!},\end{array}$(2) where $R\left(t\right)$, on the real line is convergent with Taylors' expansion given by (3) $\begin{array}{r}R\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}{R}_k\frac{t^k}{k!},R_0\ne 0.\end{array}$(3) Recently, the 2D Appell polynomials were given by Khan and Raza [2] by the generating relation: (4) $\begin{array}{r}\sum _{l=0}^{\mathrm{\infty }}{}_H{R}_l^{\left(j\right)}(p,q)\frac{t^l}{l!}=R\left(t\right)\exp (pt+q{t}^j),\end{array}$(4) where R(t) is given by (3(3) $\begin{array}{r}R\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}{R}_k\frac{t^k}{k!},R_0\ne 0.\end{array}$(3) ).

The expression (4(4) $\begin{array}{r}\sum _{l=0}^{\mathrm{\infty }}{}_H{R}_l^{\left(j\right)}(p,q)\frac{t^l}{l!}=R\left(t\right)\exp (pt+q{t}^j),\end{array}$(4) ) is a solution of the heat equation: (5) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }q}{\{}_H{R}_l^{\left(j\right)}(p,q)\}=\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{p}^j}{\{}_H{R}_l^{\left(j\right)}(p,q\left)\right\}\\ \text{and}& \\ & {}_H{R}_l^{\left(j\right)}(p,0)=R_l\left(p\right).\end{array}\end{array}$(5) In recent years, a number of generalizations of mathematical physics especially, special functions have seen a considerable evolution. The new advancement in the special functions theory provides the analytical basis for the solution of numerous mathematical physics problems, which have several wide-ranging applications. The significant advancement in the theory of generalized special functions is based on the introduction of multi-variable and multi-index special functions. The significance of special functions has been acknowledged in both pure mathematics and practical contexts. The need for multi- variable and multi-index special functions are realized to tackle the issues emerging in the theory of abstract algebra and partial differential equations. Hermite himself [3] first devised the notion of multiple-index, multiple-variable Hermite polynomials. The Hermite polynomials are found in physics, where they generate the eigenstates of the quantum harmonic oscillator and also appear in the solution of the Schrodinger equation for the harmonic oscillator. They are also used in the numerical analysis as Gaussian quadrature.

The origins of monomiality can be traced to 1941 when Steffenson developed the poweroid notion [4], which was later refined by Dattoli [5]. The $\hat{\mathcal{M}}$ and $\hat{\mathcal{D}}$ operators exist and function as multiplicative and derivative operators for a polynomial set $\left\{b_m\right(u){\}}_{m\in \mathbb{N}}$, which means that they satisfy the following expressions: (6) $\begin{array}{r}{b}_{m+1}\left(u\right)=\hat{\mathcal{M}}\left\{b_m\right(u\left)\right\}\end{array}$(6) and (7) $\begin{array}{r}m{b}_{m-1}\left(u\right)=\hat{\mathcal{D}}\left\{b_m\right(u\left)\right\}.\end{array}$(7) Then, the set $\left\{b_m\right(u){\}}_{m\in \mathbb{N}}$ manipulated by multiplicative and derivative operators is referred to as a quasi-monomial and is required to obey the formula: (8) $\begin{array}{r}[\hat{\mathcal{D}},\hat{\mathcal{M}}]=\hat{\mathcal{D}}\hat{\mathcal{M}}-\hat{\mathcal{M}}\hat{\mathcal{D}}=\hat{1},\end{array}$(8) thus displays a Weyl group structure.

The properties of the operators $\hat{\mathcal{M}}$ and $\hat{\mathcal{D}}$ can be used to determine the properties of the underlying set $\left\{b_m\right(u){\}}_{m\in \mathbb{N}}$ when it is quasi-monomial. Thus, the following traits are accurate:

1. $b_m\left(u\right)$ demonstrate the differential equation (9) $\begin{array}{r}\hat{\mathcal{M}}\hat{\mathcal{D}}\left\{b_m\right(u\left)\right\}=m{b}_m\left(u\right),\end{array}$(9) if $\hat{\mathcal{M}}$ and $\hat{\mathcal{D}}$ possesses differential realizations.

2. The explicit form of $b_m\left(u\right)$, can be cast in the form as listed: (10) $\begin{array}{r}{b}_m\left(u\right)={\hat{\mathcal{M}}}^m\left\{1\right\},\end{array}$(10) while taking, $b_0\left(u\right)=1$.

3. Also, generating relation in exponential form for $b_m\left(u\right)$ can be casted in the form (11) $\begin{array}{r}{\text{e}}^{t\hat{\mathcal{M}}}\left\{1\right\}=\sum _{m=0}^{\mathrm{\infty }}{b}_m\left(u\right)\frac{t^m}{m!},|t|<\mathrm{\infty },\end{array}$(11) by usage of identity (10(10) $\begin{array}{r}{b}_m\left(u\right)={\hat{\mathcal{M}}}^m\left\{1\right\},\end{array}$(10) ).

Recently, Hwang and Ryoo [6] introduced a 2-variable degenerate form of Hermite polynomials given by the expression listed as: (12) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{Y}}_m(u,v;\lambda )\frac{t^m}{m!}=(1+\lambda {)}^{t\left(\frac{u+vt}{\lambda }\right)},\end{array}$(12) are the solutions of the equation (13) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }v}\left\{{\mathcal{Y}}_m\right(u,v;\lambda \left)\right\}=\frac{\lambda }{\log }(1+\lambda )\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{u}^2}\left\{{\mathcal{Y}}_m\right(u,v;\lambda \left)\right\}\\ \text{and}& \\ & {\mathcal{Y}}_m(u,0;\lambda )={\left(\frac{\log (1+\lambda )}{\lambda }\right)}^m{u}^m.\end{array}\end{array}$(13) For, $\lambda \to 0$, the expression (12(12) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{Y}}_m(u,v;\lambda )\frac{t^m}{m!}=(1+\lambda {)}^{t\left(\frac{u+vt}{\lambda }\right)},\end{array}$(12) ) reduces to Equations (4(4) $\begin{array}{r}\sum _{l=0}^{\mathrm{\infty }}{}_H{R}_l^{\left(j\right)}(p,q)\frac{t^l}{l!}=R\left(t\right)\exp (pt+q{t}^j),\end{array}$(4) ) and (13(13) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }v}\left\{{\mathcal{Y}}_m\right(u,v;\lambda \left)\right\}=\frac{\lambda }{\log }(1+\lambda )\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{u}^2}\left\{{\mathcal{Y}}_m\right(u,v;\lambda \left)\right\}\\ \text{and}& \\ & {\mathcal{Y}}_m(u,0;\lambda )={\left(\frac{\log (1+\lambda )}{\lambda }\right)}^m{u}^m.\end{array}\end{array}$(13) ) reduces to (5(5) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }q}{\{}_H{R}_l^{\left(j\right)}(p,q)\}=\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{p}^j}{\{}_H{R}_l^{\left(j\right)}(p,q\left)\right\}\\ \text{and}& \\ & {}_H{R}_l^{\left(j\right)}(p,0)=R_l\left(p\right).\end{array}\end{array}$(5) ).

Moreover, the formulae for Stirling numbers of the first and second kinds are provided by Gould [7], Abramowitz and Stegun [8]: (14) $\begin{array}{r}(u{)}_l=\sum _{k=0}^l{S}_1(l,k)u^k\text{and}{u}^l=\sum _{k=0}^l{S}_2(l,k\left)\right(u{)}_k,\end{array}$(14) respectively, where $(u{)}_k=u(u-1\left)\right(u-2)\cdots (u-l+1)$ denotes falling factorial of order l.

Based on these results and motivated by them, we construct degenerate 2D bivariate Appell polynomials and establish their monomiality principle, deriving their explicit forms and implicit forms. Further, the rest part of the article is written as follows: In Section 2 we construct degenerate 2D bivariate Appell polynomials and obtain some of their significant and basic properties. In Section 3, quasi-monomial characteristics for these polynomials are established. In Section 5, a few members of this polynomial family are established and their related findings are found in the last section.

2. Degenerate 2D bivariate Appell polynomials

This section examines a brand-new class of degenerate 2D bivariate Appell polynomials. Additionally, several characteristics of these polynomials' are also obtained. Thus, using the generating function concept, we define the degenerate 2D bivariate Appell polynomials as: (15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) or (16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) For, $v\to 0$, the degenerate 2D bivariate Appell polynomials reduce to degenerate Appell polynomials [9] given by (17) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m(u;\lambda )\frac{t^m}{m!}.\end{array}$(17) Further, for $(1+\lambda {)}^{\frac{ut}{\lambda }}\to \exp (ut)$, it is evident that (17(17) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m(u;\lambda )\frac{t^m}{m!}.\end{array}$(17) ) reduces to (2(2) $\begin{array}{r}R\left(t\right)\exp \left(ut\right)=\sum _{k=0}^{\mathrm{\infty }}{R}_k\left(u\right)\frac{t^k}{k!},\end{array}$(2) ).

Also, for $(1+\lambda {)}^{\frac{ut}{\lambda }}\to \exp (ut)$ and $(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}\to \exp (v{t}^j)$, it is clear that (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) (or (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) )) reduces to (4(4) $\begin{array}{r}\sum _{l=0}^{\mathrm{\infty }}{}_H{R}_l^{\left(j\right)}(p,q)\frac{t^l}{l!}=R\left(t\right)\exp (pt+q{t}^j),\end{array}$(4) ).

Next, we derive the explicit forms of the degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ by proving the following result:

Theorem 2.1

The degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ satisfy the following explicit form: (18) $\begin{array}{r}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )=m!\sum _{l=0}^{\left[\frac{m}{j}\right]}{\mathcal{R}}_{m-jl}(u;\lambda ){\left(\frac{\log (1+\lambda )}{\lambda }\right)}^l\frac{v^l}{l!(m-jl)!}.\end{array}$(18)

Proof.

Inserting expression (17(17) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m(u;\lambda )\frac{t^m}{m!}.\end{array}$(17) ) and expression (19) $\begin{array}{r}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{l=0}^{\mathrm{\infty }}{\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{jl}}{l!}\end{array}$(19) in generating relation (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) (or (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) )), it follows that (20) $\begin{array}{r}\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m(u;\lambda )\frac{t^m}{m!}\sum _{l=0}^{\mathrm{\infty }}{\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{jl}}{l!}.\end{array}$(20) Further simplifying above expression, we find (21) $\begin{array}{r}\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}{\mathcal{R}}_m(u;\lambda ){\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{m+jl}}{m!l!}.\end{array}$(21) Using (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) in the l.h.s. of Equation (21(21) $\begin{array}{r}\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}{\mathcal{R}}_m(u;\lambda ){\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{m+jl}}{m!l!}.\end{array}$(21) ) and replacing m by m−jl, it follows that $\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}{\mathcal{R}}_{m-jl}(u;\lambda ){\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^m}{(m-jl)!l!}.$ Therefore on using C.P. (Cauchy Product) rule in the r.h.s. of the previous equation, assertion (18(18) $\begin{array}{r}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )=m!\sum _{l=0}^{\left[\frac{m}{j}\right]}{\mathcal{R}}_{m-jl}(u;\lambda ){\left(\frac{\log (1+\lambda )}{\lambda }\right)}^l\frac{v^l}{l!(m-jl)!}.\end{array}$(18) ) is proved by comparing coefficients of the same powers of t on both sides of the resultant equation.

Remark 2.1

For, $\lambda \to 0$ in (18(18) $\begin{array}{r}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )=m!\sum _{l=0}^{\left[\frac{m}{j}\right]}{\mathcal{R}}_{m-jl}(u;\lambda ){\left(\frac{\log (1+\lambda )}{\lambda }\right)}^l\frac{v^l}{l!(m-jl)!}.\end{array}$(18) ), we find the explicit form satisfied by Equation (4(4) $\begin{array}{r}\sum _{l=0}^{\mathrm{\infty }}{}_H{R}_l^{\left(j\right)}(p,q)\frac{t^l}{l!}=R\left(t\right)\exp (pt+q{t}^j),\end{array}$(4) ).

Further, we find explicit forms of the degenerate 2D bivariate Appell polynomials, by proving the results listed below:

Theorem 2.2

The degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ satisfy the following explicit form: (22) $\begin{array}{r}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )=m!\sum _{k=0}^{\left[m\right]}\sum _{l=0}^{\left[\frac{k}{j}\right]}{\mathcal{R}}_{m-k}{\left(\frac{\log (1+\lambda )}{\lambda }\right)}^k{\left(\frac{\log (1+\lambda )}{\lambda }\right)}^l\frac{u^k{v}^l}{(m-k)!(k-jl)!l!}.\end{array}$(22)

Proof.

Inserting expressions (3(3) $\begin{array}{r}R\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}{R}_k\frac{t^k}{k!},R_0\ne 0.\end{array}$(3) ), (19(19) $\begin{array}{r}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{l=0}^{\mathrm{\infty }}{\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{jl}}{l!}\end{array}$(19) ) and expression (23) $\begin{array}{r}(1+\lambda {)}^{\frac{ut}{\lambda }}=\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^k\frac{t^k}{k!}\end{array}$(23) in generating relation (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) (or (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) )), it follows that (24) $\begin{array}{rl}& \mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}\\ & =\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m\frac{t^m}{m!}\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^k\frac{t^k}{k!}\sum _{l=0}^{\mathrm{\infty }}{\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{jl}}{l!}.\end{array}$(24) Further, simplifying the above expression, we have (25) $\begin{array}{r}\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m\frac{t^m}{m!}\sum _{k=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^k{\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{k+jl}}{k!l!}.\end{array}$(25) Replacing k by k−jl and using C.P. (Cauchy Product) rule in the r.h.s. of previous equation, it follows that (26) $\begin{array}{rl}& \mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}\\ & =\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m\sum _{k=0}^{\mathrm{\infty }}\sum _{l=0}^{\left[\frac{k}{j}\right]}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^k{\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{m+k}}{m!(k-jl)!l!}.\end{array}$(26) Therefore, using (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) in the l.h.s. of previous equation and replacing m by m−k in the r.h.s. of previous equation, it follows that (27) $\begin{array}{rl}& \sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}\\ & =\sum _{m=0}^{\mathrm{\infty }}\sum _{k=0}^{\left[m\right]}\sum _{l=0}^{\left[\frac{k}{j}\right]}{\mathcal{R}}_{m-k}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^k{\left(\frac{v\log (1+\lambda )}{\lambda }\right)}^l\frac{t^m}{(m-k)!(k-jl)!l!}.\end{array}$(27) Thus, equating the coefficients of same powers of t on both sides, assertion (22(22) $\begin{array}{r}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )=m!\sum _{k=0}^{\left[m\right]}\sum _{l=0}^{\left[\frac{k}{j}\right]}{\mathcal{R}}_{m-k}{\left(\frac{\log (1+\lambda )}{\lambda }\right)}^k{\left(\frac{\log (1+\lambda )}{\lambda }\right)}^l\frac{u^k{v}^l}{(m-k)!(k-jl)!l!}.\end{array}$(22) ) is proved.

Theorem 2.3

The degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ satisfy the following explicit form: (28) $\begin{array}{r}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )=m!\sum _{l=0}^{\left[m\right]}{\mathcal{R}}_{m-l}^{\left[j\right]}(v;\lambda ){\left(\frac{\log (1+\lambda )}{\lambda }\right)}^l\frac{u^l}{l!(m-l)!}.\end{array}$(28)

Proof.

Inserting expression (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) with v = 0 and expression (23(23) $\begin{array}{r}(1+\lambda {)}^{\frac{ut}{\lambda }}=\sum _{k=0}^{\mathrm{\infty }}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^k\frac{t^k}{k!}\end{array}$(23) ) in generating relation (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) ), it follows that (29) $\begin{array}{r}\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m^{\left[j\right]}(v;\lambda )\sum _{l=0}^{\mathrm{\infty }}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^l\frac{t^{m+l}}{m!l!}.\end{array}$(29) Using (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) in the l.h.s. of the previous equation and replacing m by m−l and using C.P. (Cauchy Product) rule in the r.h.s. of the previous equation, it follows that $\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m^{\left[j\right]}(u,v;\lambda )=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_{m-l}^{\left[j\right]}(v;\lambda )\sum _{l=0}^{\left[m\right]}{\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^l\frac{t^m}{(m-l)!l!},$ which further can be written as $\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m^{\left[j\right]}(u,v;\lambda )=\sum _{m=0}^{\mathrm{\infty }}\sum _{l=0}^{\left[m\right]}(\genfrac{}{}{0ex}{}{m}{l}){\mathcal{R}}_{m-l}^{\left[j\right]}(v;\lambda ){\left(\frac{u\log (1+\lambda )}{\lambda }\right)}^l\frac{t^m}{m!}.$ Assertion (28(28) $\begin{array}{r}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )=m!\sum _{l=0}^{\left[m\right]}{\mathcal{R}}_{m-l}^{\left[j\right]}(v;\lambda ){\left(\frac{\log (1+\lambda )}{\lambda }\right)}^l\frac{u^l}{l!(m-l)!}.\end{array}$(28) ) is proved by comparing coefficients of same powers of t on both sides of the previous equation.

In the next section, we obtain the quasi-monomial properties for the degenerate 2D bivariate Appell polynomials.

3. Quasi-monomial properties and summation formulae

Here, we establish the quasi-monomial properties for the degenerate 2D bivariate Appell polynomials, by proving the following results:

Theorem 3.1

For, the degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$, the following multiplicative and derivative operator holds true: (30) $\begin{array}{r}{\mathcal{G}}_{m+1}^{\left[j\right]}(u,v;\lambda )=\stackrel{ˆ}{{\mathcal{M}}_{{\mathcal{G}}^{\left[j\right]}}}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=(\frac{u}{\eta }+\frac{{\mathcal{R}}^{\prime }\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}{\mathcal{R}\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}+jv{\eta }^{j-2}\frac{{\mathrm{\partial }}^{j-1}}{\mathrm{\partial }{u}^{j-1}})\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}\end{array}$(30) and (31) $\begin{array}{r}m{\mathcal{G}}_{m-1}^{\left[j\right]}(u,v;\lambda )=\stackrel{ˆ}{{\mathcal{D}}_{{\mathcal{G}}^{\left[j\right]}}}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}.\end{array}$(31)

Proof.

Taking the derivatives of generating expression (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) (or (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) )) w.r.t. t, we find (32) $\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\}& =(\frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}+\frac{\log (1+\lambda )}{\lambda }(u+jv{t}^{j-1}))\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\}\\ & =m{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^{m-1}}{m!}.\end{array}$(32) Making use of identity (33) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\}=\left(\frac{\log (1+\lambda )}{\lambda }t\right)\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\},\\ & \text{inserting Equation~(2.1) in above equation, it follows that}\\ & \frac{\lambda }{\log (1+\lambda )}\frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\}=t\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\},\end{array}\end{array}$(33) taking $\frac{\lambda }{\log (1+\lambda )}=\eta$ in (33(33) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\}=\left(\frac{\log (1+\lambda )}{\lambda }t\right)\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\},\\ & \text{inserting Equation~(2.1) in above equation, it follows that}\\ & \frac{\lambda }{\log (1+\lambda )}\frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\}=t\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\},\end{array}\end{array}$(33) ) and using Equation (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) in Equation (32(32) $\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\}& =(\frac{{\mathcal{R}}^{\prime }\left(t\right)}{\mathcal{R}\left(t\right)}+\frac{\log (1+\lambda )}{\lambda }(u+jv{t}^{j-1}))\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\}\\ & =m{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^{m-1}}{m!}.\end{array}$(32) ) with m replaced by m + 1 in last summation, it follows that (34) $\begin{array}{rl}& (\frac{{\mathcal{R}}^{\prime }\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}{\mathcal{R}\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}+\frac{\log (1+\lambda )}{\lambda }(u+jv{\eta }^{j-1}\frac{{\mathrm{\partial }}^{j-1}}{\mathrm{\partial }{u}^{j-1}}))\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\}\\ & ={\mathcal{G}}_{m+1}^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}.\end{array}$(34) Comparing the coefficients of like powers of $\frac{t^m}{m!}$ in previous equation and in view of (6(6) $\begin{array}{r}{b}_{m+1}\left(u\right)=\hat{\mathcal{M}}\left\{b_m\right(u\left)\right\}\end{array}$(6) ), assertion (30(30) $\begin{array}{r}{\mathcal{G}}_{m+1}^{\left[j\right]}(u,v;\lambda )=\stackrel{ˆ}{{\mathcal{M}}_{{\mathcal{G}}^{\left[j\right]}}}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=(\frac{u}{\eta }+\frac{{\mathcal{R}}^{\prime }\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}{\mathcal{R}\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}+jv{\eta }^{j-2}\frac{{\mathrm{\partial }}^{j-1}}{\mathrm{\partial }{u}^{j-1}})\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}\end{array}$(30) ) is proved.

Also, in view of identity (33(33) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\}=\left(\frac{\log (1+\lambda )}{\lambda }t\right)\left\{\mathcal{R}\right(t\left)\right(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)}\},\\ & \text{inserting Equation~(2.1) in above equation, it follows that}\\ & \frac{\lambda }{\log (1+\lambda )}\frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\}=t\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\},\end{array}\end{array}$(33) ), we find (35) $\begin{array}{r}\frac{\lambda }{\log (1+\lambda )}\frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\}=\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^{m+1}}{m!}\right\},\end{array}$(35) replacing m by m−1 in r.h.s. of the previous equation, it follows that $\frac{\lambda }{\log (1+\lambda )}\frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\}=m\left\{{\mathcal{G}}_{m-1}^{\left[j\right]}\right(u,v;\lambda \left)\frac{t^m}{m!}\right\}.$ Thus, comparing the coefficients of the same powers of t on both sides of the previous equation, assertion (31(31) $\begin{array}{r}m{\mathcal{G}}_{m-1}^{\left[j\right]}(u,v;\lambda )=\stackrel{ˆ}{{\mathcal{D}}_{{\mathcal{G}}^{\left[j\right]}}}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}.\end{array}$(31) ) is proved.

Next, we find the differential equation satisfied by degenerate 2D bivariate Appell polynomials by proving the following result:

Theorem 3.2

For, degenerate 2D bivariate Appell polynomials, the following differential equation holds true: (36) $\begin{array}{r}((u+\eta \frac{{\mathcal{R}}^{\prime }\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}{\mathcal{R}\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)})\frac{\mathrm{\partial }}{\mathrm{\partial }u}+jv{\eta }^{j-1}\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{u}^j}-m)\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=0.\end{array}$(36)

Proof.

Inserting Equations (30(30) $\begin{array}{r}{\mathcal{G}}_{m+1}^{\left[j\right]}(u,v;\lambda )=\stackrel{ˆ}{{\mathcal{M}}_{{\mathcal{G}}^{\left[j\right]}}}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=(\frac{u}{\eta }+\frac{{\mathcal{R}}^{\prime }\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}{\mathcal{R}\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}+jv{\eta }^{j-2}\frac{{\mathrm{\partial }}^{j-1}}{\mathrm{\partial }{u}^{j-1}})\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}\end{array}$(30) ) and (31(31) $\begin{array}{r}m{\mathcal{G}}_{m-1}^{\left[j\right]}(u,v;\lambda )=\stackrel{ˆ}{{\mathcal{D}}_{{\mathcal{G}}^{\left[j\right]}}}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}.\end{array}$(31) ) in Equation (9(9) $\begin{array}{r}\hat{\mathcal{M}}\hat{\mathcal{D}}\left\{b_m\right(u\left)\right\}=m{b}_m\left(u\right),\end{array}$(9) ), it follows that $(\frac{u}{\eta }+\frac{{\mathcal{R}}^{\prime }\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}{\mathcal{R}\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}+jv{\eta }^{j-2}\frac{{\mathrm{\partial }}^{j-1}}{\mathrm{\partial }{u}^{j-1}})\eta \left(\frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=m\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}.$ Therefore, upon simplification, assertion (36(36) $\begin{array}{r}((u+\eta \frac{{\mathcal{R}}^{\prime }\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)}{\mathcal{R}\left(\eta \frac{\mathrm{\partial }}{\mathrm{\partial }u}\right)})\frac{\mathrm{\partial }}{\mathrm{\partial }u}+jv{\eta }^{j-1}\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{u}^j}-m)\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=0.\end{array}$(36) ) is proved.

Next, we derive the summation formulae for the degenerate 2D bivariate Appell polynomials in terms of Stirling numbers of the first kind by proving the following results:

Theorem 3.3

The degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ satisfy the following summation formulae in terms of Stirling numbers of the first kind: (37) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{R}}_{m-jq}(u;\lambda ),\end{array}$(37) (38) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{G}}_{m-p}^{\left[j\right]}(v;\lambda ),\end{array}$(38) (39) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v_1+v_2;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v_2^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{G}}_{m-jq}^{\left[j\right]}(u,v_1;\lambda ),\end{array}$(39) (40) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u_1+u_2,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u_2^p{\lambda }^{k-p}\frac{(p!}{k!}{\mathcal{G}}_{m-p}^{\left[j\right]}(u_1,v;\lambda ),\end{array}$(40) (41) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(0,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{R}}_{m-jq}(0;\lambda ),\end{array}$(41) (42) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{R}}_{m-p}(0;\lambda ),\end{array}$(42) respectively.

Proof.

Using expressions (17(17) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m(u;\lambda )\frac{t^m}{m!}.\end{array}$(17) ) and (14(14) $\begin{array}{r}(u{)}_l=\sum _{k=0}^l{S}_1(l,k)u^k\text{and}{u}^l=\sum _{k=0}^l{S}_2(l,k\left)\right(u{)}_k,\end{array}$(14) ) in generating relation (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ), assertion (37(37) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{R}}_{m-jq}(u;\lambda ),\end{array}$(37) ) is proved.

Similarly, using Equations (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) with u = 0 and (14(14) $\begin{array}{r}(u{)}_l=\sum _{k=0}^l{S}_1(l,k)u^k\text{and}{u}^l=\sum _{k=0}^l{S}_2(l,k\left)\right(u{)}_k,\end{array}$(14) ), assertion (38(38) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{G}}_{m-p}^{\left[j\right]}(v;\lambda ),\end{array}$(38) ) is proved.

Further, replacing $u\to u_1+u_2$ in Equation (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) and using (14(14) $\begin{array}{r}(u{)}_l=\sum _{k=0}^l{S}_1(l,k)u^k\text{and}{u}^l=\sum _{k=0}^l{S}_2(l,k\left)\right(u{)}_k,\end{array}$(14) ), assertion (39(39) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v_1+v_2;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v_2^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{G}}_{m-jq}^{\left[j\right]}(u,v_1;\lambda ),\end{array}$(39) ) is proved.

Further more, replacing $v\to v_1+v_2$ in Equation (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) and using (14(14) $\begin{array}{r}(u{)}_l=\sum _{k=0}^l{S}_1(l,k)u^k\text{and}{u}^l=\sum _{k=0}^l{S}_2(l,k\left)\right(u{)}_k,\end{array}$(14) ), assertion (40(40) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u_1+u_2,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u_2^p{\lambda }^{k-p}\frac{(p!}{k!}{\mathcal{G}}_{m-p}^{\left[j\right]}(u_1,v;\lambda ),\end{array}$(40) ) is proved.

Again, Equations (41(41) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(0,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{R}}_{m-jq}(0;\lambda ),\end{array}$(41) ) and (42(42) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{R}}_{m-p}(0;\lambda ),\end{array}$(42) ) are obvious in view of Equations (37(37) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{R}}_{m-jq}(u;\lambda ),\end{array}$(37) ) and (38(38) $\begin{array}{rl}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{G}}_{m-p}^{\left[j\right]}(v;\lambda ),\end{array}$(38) ).

In the next section, we will give operational formalism to the degenerate 2D bivariate Appell polynomials and then find their determinant form.

4. Operational rules and determinant forms

These operational approaches are still used today in many areas of mathematical physics, quantum mechanics, and classical optics. Therefore, these techniques provide effective and potent tools of research, see for example [10–25].

As, defined by the generating relation (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) (or (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) )), the degenerate 2D bivariate polynomials polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ are solutions' of the equation: (43) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }v}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=\frac{\lambda }{\log (1+\lambda )}\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{u}^j}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}\\ \text{and}& \\ & {\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )={\mathcal{R}}_m(u;\lambda )\end{array}\end{array}$(43) respectively.

Thus, in view of Equations (43(43) $\begin{array}{r}\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }v}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}=\frac{\lambda }{\log (1+\lambda )}\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{u}^j}\left\{{\mathcal{G}}_m^{\left[j\right]}\right(u,v;\lambda \left)\right\}\\ \text{and}& \\ & {\mathcal{G}}_m^{\left[j\right]}(u,0;\lambda )={\mathcal{R}}_m(u;\lambda )\end{array}\end{array}$(43) ) and (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ) (or (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) )), it follows that the degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ can be expressed as: (44) $\begin{array}{r}\exp \left(v\frac{\lambda }{\log (1+\lambda )}\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{u}^j}\right){\mathcal{R}}_m(u;\lambda )={\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda ),\end{array}$(44) or (45) $\begin{array}{r}\exp \left(\eta v\frac{{\mathrm{\partial }}^j}{\mathrm{\partial }{u}^j}\right){\mathcal{R}}_m(u;\lambda )={\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda ),\end{array}$(45) respectively.

Next, we establish the determinant form of the degenerate 2D bivariate Appell polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$, by taking into consideration the following form of Equation (16(16) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\frac{ut}{\lambda }}(1+\lambda {)}^{\frac{v{t}^j}{\lambda }}.\end{array}$(16) ): (46) $\begin{array}{r}\mathcal{R}\left(t\right)\sum _{m=0}^{\mathrm{\infty }}{\mathcal{Y}}_m^{\left[j\right]}(u,v;\lambda )=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda ),\end{array}$(46) where (47) $\begin{array}{r}\mathcal{R}\left(t\right)=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{R}}_m\frac{t^m}{m!},{\mathcal{R}}_0\ne 0.\end{array}$(47) Thus, in view of the above expressions, the polynomials ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )$ satisfy the following determinant form:

Theorem 4.1

The degenerate 2D bivariate Appell polynomials satisfy the following determinant: (48) $\begin{array}{rl}& {\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\\ & =\frac{(-1{)}^n}{{\left({\gamma }_0\right)}^{n+1}}\left|\begin{array}{cccccc}1& {\mathcal{Y}}_1^{\left[j\right]}(u,v;\lambda )& {\mathcal{Y}}_2^{\left[j\right]}(u,v;\lambda )& \cdots & {\mathcal{Y}}_{m-1}^{\left[j\right]}(u,v;\lambda )& {\mathcal{Y}}_m^{\left[j\right]}(u,v;\lambda )\\ {\gamma }_0& {\gamma }_1& {\gamma }_2& \cdots & {\gamma }_{m-1}& {\gamma }_m\\ 0& {\gamma }_0& (\genfrac{}{}{0ex}{}{2}{1}){\gamma }_1& \cdots & (\genfrac{}{}{0ex}{}{m-1}{1}){\gamma }_{m-2}& (\genfrac{}{}{0ex}{}{m}{1}){\gamma }_{m-1}\\ 0& 0& {\gamma }_0& \cdots & (\genfrac{}{}{0ex}{}{m-1}{2}){\gamma }_{m-3}& (\genfrac{}{}{0ex}{}{m}{2}){\gamma }_{m-2}\\ .& .& .& \cdots & .& .\\ .& .& .& \cdots & .& .\\ 0& 0& 0& \cdots & {\gamma }_0& (\genfrac{}{}{0ex}{}{m}{m-1}){\gamma }_1\end{array}\right|,\end{array}$(48) where ${\gamma }_m,m=0,1,\dots \text{are the coefficients of Maclaurin's series}\frac{1}{\mathcal{R}\left(t\right)}$

Proof.

Multiplying both sides of Equation (46(46) $\begin{array}{r}\mathcal{R}\left(t\right)\sum _{m=0}^{\mathrm{\infty }}{\mathcal{Y}}_m^{\left[j\right]}(u,v;\lambda )=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda ),\end{array}$(46) ) by $\frac{1}{\mathcal{R}\left(t\right)}=\sum _{k=0}^{\mathrm{\infty }}{\gamma }_k\frac{t^k}{k!}$, we find (49) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{Y}}_m^{\left[j\right]}(u,v;\lambda )=\sum _{k=0}^{\mathrm{\infty }}{\gamma }_k\frac{t^k}{k!}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda ).\end{array}$(49) By using C.P. rule in Equation (49(49) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{Y}}_m^{\left[j\right]}(u,v;\lambda )=\sum _{k=0}^{\mathrm{\infty }}{\gamma }_k\frac{t^k}{k!}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda ).\end{array}$(49) ), we have (50) $\begin{array}{r}{\mathcal{Y}}_m^{\left[j\right]}(u,v;\lambda )=\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{k}){\gamma }_k{\mathcal{G}}_{m-K}^{\left[j\right]}(u,v;\lambda ).\end{array}$(50) The system of $m-$ equations with unknowns ${\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda ),m=0,1,2,\dots$ is produced as a result of this equality. Cramers' rule is used to solve this system and using the fact that the denominator is the determinant of the lower triangular matrix with determinant $({\gamma }_0{)}^{m+1}$ and by taking transpose of the numerator, then replacing the ith row by $(i+1)$th position, for $i=1,2,\dots m-1$ gives the desired result.

5. Examples

A variety of members of the Appell polynomial family can be obtained depending on the proper choice for the function $\mathcal{R}\left(t\right)$. Numerous applications in number theory, combinatorics, numerical analysis, and other areas of practical mathematics make use of these polynomials and numbers of Bernoulli, Euler, and Genocchi. The Taylor expansion, the trigonometric and hyperbolic tangent and cotangent functions, and the sums of powers of natural numbers are only a few examples of mathematical formulas where the Bernoulli numbers can be found. In close proximity to the trigonometric and hyperbolic secant function origins, the Euler numbers enter the Taylor expansion. In graph theory, automata theory, and calculating the number of up-down ascending sequences, the Genocchi numbers are useful.

Thus for suitable selection of $\mathcal{(}R\left)\right(t)$ in (15(15) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\mathcal{R}\left(t\right)(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(15) ), the following generating functions for degenerate 2D Bernoulli, Euler and Genocchi polynomials hold: (51) $\begin{array}{rl}& \sum _{m=0}^{\mathrm{\infty }}{\mathcal{B}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\frac{t}{{\text{e}}^t-1}(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(51) (52) $\begin{array}{rl}& \sum _{m=0}^{\mathrm{\infty }}{\mathcal{E}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\frac{2}{{\text{e}}^t+1}(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(52) and (53) $\begin{array}{r}\sum _{m=0}^{\mathrm{\infty }}{\mathcal{G}\mathcal{E}}_m^{\left[j\right]}(u,v;\lambda )\frac{t^m}{m!}=\frac{2t}{{\text{e}}^t+1}(1+\lambda {)}^{\left(\frac{ut+v{t}^j}{\lambda }\right)},\end{array}$(53) respectively. Thus the corresponding results can be obtained for these polynomials.

Theorem 5.1

The degenerate 2D bivariate Bernoulli, Euler and Genocchi polynomials satisfy the following summation formulae in terms of Stirling numbers of the first kind: (54) $\begin{array}{rl}{\mathcal{B}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{B}}_{m-jq}(u;\lambda ),\end{array}$(54) (55) $\begin{array}{rl}{\mathcal{B}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{B}}_{m-p}^{\left[j\right]}(v;\lambda ),\end{array}$(55) (56) $\begin{array}{rl}{\mathcal{B}}_m^{\left[j\right]}(u,v_1+v_2;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v_2^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{B}}_{m-jq}^{\left[j\right]}(u,v_1;\lambda ),\end{array}$(56) (57) $\begin{array}{rl}{\mathcal{B}}_m^{\left[j\right]}(u_1+u_2,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u_2^p{\lambda }^{k-p}\frac{(p!}{k!}{\mathcal{B}}_{m-p}^{\left[j\right]}(u_1,v;\lambda ),\end{array}$(57) (58) $\begin{array}{rl}{\mathcal{B}}_m^{\left[j\right]}(0,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{B}}_{m-jq}(0;\lambda ),\end{array}$(58) (59) $\begin{array}{rl}{\mathcal{B}}_m^{\left[j\right]}(u,0;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{B}}_{m-p}(0;\lambda ),\end{array}$(59) (60) $\begin{array}{rl}{\mathcal{E}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{E}}_{m-jq}(u;\lambda ),\end{array}$(60) (61) $\begin{array}{rl}{\mathcal{E}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{E}}_{m-p}^{\left[j\right]}(v;\lambda ),\end{array}$(61) (62) $\begin{array}{rl}{\mathcal{E}}_m^{\left[j\right]}(u,v_1+v_2;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v_2^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{E}}_{m-jq}^{\left[j\right]}(u,v_1;\lambda ),\end{array}$(62) (63) $\begin{array}{rl}{\mathcal{E}}_m^{\left[j\right]}(u_1+u_2,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u_2^p{\lambda }^{k-p}\frac{(p!}{k!}{\mathcal{E}}_{m-p}^{\left[j\right]}(u_1,v;\lambda ),\end{array}$(63) (64) $\begin{array}{rl}{\mathcal{E}}_m^{\left[j\right]}(0,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{E}}_{m-jq}(0;\lambda ),\end{array}$(64) (65) $\begin{array}{rl}{\mathcal{E}}_m^{\left[j\right]}(u,0;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{E}}_{m-p}(0;\lambda ),\end{array}$(65) and (66) $\begin{array}{rl}{\mathcal{G}\mathcal{E}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{G}\mathcal{E}}_{m-jq}(u;\lambda ),\end{array}$(66) (67) $\begin{array}{rl}{\mathcal{G}\mathcal{E}}_m^{\left[j\right]}(u,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{G}\mathcal{E}}_{m-p}^{\left[j\right]}(v;\lambda ),\end{array}$(67) (68) $\begin{array}{rl}{\mathcal{G}\mathcal{E}}_m^{\left[j\right]}(u,v_1+v_2;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v_2^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{G}\mathcal{E}}_{m-jq}^{\left[j\right]}(u,v_1;\lambda ),\end{array}$(68) (69) $\begin{array}{rl}{\mathcal{G}\mathcal{E}}_m^{\left[j\right]}(u_1+u_2,v;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u_2^p{\lambda }^{k-p}\frac{(p!}{k!}{\mathcal{G}\mathcal{E}}_{m-p}^{\left[j\right]}(u_1,v;\lambda ),\end{array}$(69) (70) $\begin{array}{rl}{\mathcal{G}\mathcal{E}}_m^{\left[j\right]}(0,v;\lambda )& =\sum _{q=0}^{\left[\frac{m}{j}\right]}\sum _{i=q}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{jq})S_1(i,q)v^q{\lambda }^{i-q}\frac{\left(jq\right)!}{i!}{\mathcal{G}\mathcal{E}}_{m-jq}(0;\lambda ),\end{array}$(70) (71) $\begin{array}{rl}{\mathcal{G}\mathcal{E}}_m^{\left[j\right]}(u,0;\lambda )& =\sum _{p=0}^{\left[m\right]}\sum _{k=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{m}{p})S_1(k,p)u^p{\lambda }^{k-p}\frac{p!}{k!}{\mathcal{G}\mathcal{E}}_{m-p}(0;\lambda ),\end{array}$(71) respectively.

Similarly, in the same fashion other corresponding results for these polynomials can be established.

Further, future investigations and observations can be used to establish extended, generalized forms, integral representations, and other properties of the above-mentioned polynomials. Also, the determinant forms and summation formulae can also be a problem for new observations.

Acknowledgments

The authors appreciate the referees' comments and recommendations, which considerably enhanced the original paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).