Sequences of twice-iterated Δ_w-Gould–Hopper Appell polynomials

ABSTRACT

In this paper, we introduce general sequence of twice-iterated ${\mathrm{\Delta }}_w$-(degenerate) Gould–Hopper Appell polynomials (TI-DGHAP) via discrete ${\mathrm{\Delta }}_w$-Gould–Hopper Appell convolution. We obtain some of their characteristic properties such as explicit representation, determinantal representation, recurrence relation, lowering operator (LO), raising operator (RO), difference equation (DE), integro-partial lowering operator (IPLO), integro-partial raising operator (IPRO) and integro-partial difference equation (IPDE). As special cases of these general polynomials, we present TI degenerate Gould–Hopper Bernoulli polynomials, TI degenerate Gould–Hopper Poisson–Charlier polynomials, TI degenerate Gould–Hopper Boole polynomials and TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials. We also state their corresponding characteristic properties.

KEYWORDS:

• ${\mathrm{\Delta }}_w$-Gould–Hopper Appell polynomials
• recurrence relation
• shift operators
• difference equations

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 33C65
• Secondary 11B83
• 11B68

1. Introduction

Families of special polynomials have been the subject of interest in mathematics due to their nice characteristic properties, which arises in the solutions of many applied problems. Among them, Appell polynomials have special importance, because of their simple and useful definition i.e. [1–7], (1) $\frac{\mathrm{d}}{\mathrm{d}x}{A}_n\left(x\right)=n{A}_{n-1}\left(x\right).$(1) Appell polynomial sequence $\left\{A_k\right(x){\}}_{k\in {\mathbb{N}}_0}$ can be equivalently defined by the generating relation $a\left(z\right){\mathrm{e}}^{xz}=\sum _{k=0}^{\mathrm{\infty }}{A}_k\left(x\right)\frac{z^k}{k!}$where $a\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}{A}_n\frac{z^n}{n!},A_0\ne 0.$These simple definitions play the key point in the investigation of their many interesting properties (see [3–5, 8–14]). Because of their wide range of applications, several variants of Appell polynomials have been intensively studied especially in the last two decades. In [6], Khan and Raza introduced 2-iterated Appell polynomials. Twice-iterated ${\mathrm{\Delta }}_w$-Appell polynomials were introduced by [15] and their various properties are included. Twice-iterated 2D q-Appell polynomials, one of the generalizations of twice-iterated Appell polynomials, were introduced in [16], and the difference equations and various properties of these polynomials are included in [17]. 2-iterated Appell polynomials have been defined by the generating relation in [6] $a_2\left(z\right)a_1\left(z\right){\mathrm{e}}^{xz}=\sum _{k=0}^{\mathrm{\infty }}{R}_k^{\left[2\right]}\left(x\right)\frac{z^k}{k!},$where $a_1\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}{\alpha }_k\frac{z^k}{k!},{\alpha }_0\ne 0$and $a_2\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}{\text{β}}_k\frac{z^k}{k!},{\text{β}}_0\ne 0.$It is clear that from [6], they satisfy $D_x\left(R_k^{\left[2\right]}\left(x\right)\right)=k{R}_{k-1}^{\left[2\right]}\left(x\right),k\in \mathbb{N}.$On the other hand, in [13], ${}_x{\mathrm{\Delta }}_w$-Appell polynomials are defined by (2) ${}_x{\mathrm{\Delta }}_w{A}_k=kw{A}_{k-1},k\in \mathbb{N},w\in {\mathbb{R}}^{+}$(2) where ${}_x{\mathrm{\Delta }}_w$ is the finite difference operator [18] given by ${}_x{\mathrm{\Delta }}_w\left[u\right]=u(x+w)-u\left(x\right).$

Finite difference operator of order s is defined in [18] as follows: $\begin{array}{rl}{}_x{\mathrm{\Delta }}_w^s\left[u\right]& ={}_x{\mathrm{\Delta }}_w\left({}_x{\mathrm{\Delta }}_w^{s-1}\left[u\right]\right)\\ & =\sum _{r=0}^s{(-1)}^{s-r}(\genfrac{}{}{0ex}{}{s}{r})u(x+rw),s\in \mathbb{N}\end{array}$and ${}_x{\mathrm{\Delta }}_w^0=I,{}_x{\mathrm{\Delta }}_w^1={}_x{\mathrm{\Delta }}_w$. An equivalent definition to (2(2) ${}_x{\mathrm{\Delta }}_w{A}_k=kw{A}_{k-1},k\in \mathbb{N},w\in {\mathbb{R}}^{+}$(2) ), from [13], can be given by the generating relation $a\left(z\right){(1+wz)}^{\frac{x}{w}}=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k\left(x\right)\frac{z^k}{k!},$where $a\left(z\right)=\sum _{j=0}^{\mathrm{\infty }}{\alpha }_{j,w}\frac{t^j}{j!},{\alpha }_{0,w}\ne 0.$From [13], the explicit representation of the ${}_x{\mathrm{\Delta }}_w$-Appell sequence is expressed as follows: $\begin{array}{rl}{\mathcal{A}}_k\left(x\right)& ={\alpha }_k+(\genfrac{}{}{0ex}{}{k}{1}){\alpha }_{k-1}{\left(x\right)}_1^w+(\genfrac{}{}{0ex}{}{k}{2}){\alpha }_{k-2}{\left(x\right)}_2^w\\ & +\cdots +{\alpha }_0{\left(x\right)}_k^w,\{k\in {\mathbb{N}}_0=\{0,1,\dots \}\}\end{array}$where ${\left(x\right)}_k^w={(-\frac{x}{w})}_k{(-w)}^k$and Pochhammer symbol is defined by [18]: $\begin{array}{rl}{\left(x\right)}_0& =1\\ {\left(x\right)}_n& =x(x+1)\dots (x+n-1),n=1,2,\dots .\end{array}$In [15], TI-${\mathrm{\Delta }}_w$-Appell polynomials were defined by the following generating relation: $a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}(x,y;w)\frac{z^k}{k!}.$On the other hand, Gould–Hopper polynomials of the rth order are given in [19–21] and has the following generating function (3) ${\mathrm{e}}^{xt+y{t}^r}=\sum _{k=0}^{\mathrm{\infty }}{g}_k^r(x,y)\frac{t^k}{k!}.$(3) If r= 2 in (3(3) ${\mathrm{e}}^{xt+y{t}^r}=\sum _{k=0}^{\mathrm{\infty }}{g}_k^r(x,y)\frac{t^k}{k!}.$(3) ), it reduces to Gould–Hopper (2-variable Hermite Kampé de Fériet) polynomials. Additionally, its relationship with Brafman polynomials in terms of hypergeometric function was examined in [22]. The generating function of the Gould–Hopper Appell polynomials given in [4,23] is as follows: $A\left(t\right){\mathrm{e}}^{xt+y{t}^2}=\sum _{k=0}^{\mathrm{\infty }}{}_H{A}_k(x,y)\frac{t^k}{k!}.$${\mathrm{\Delta }}_w$-GHA polynomials, which are a generalization of these polynomials, are defined in [24] and have the following generating function: $a\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k(x,y;w)\frac{z^k}{k!},$where $a\left(z\right)$ is given by $a\left(z\right)=\sum _{j=0}^{\mathrm{\infty }}{\alpha }_{j,w}\frac{z^j}{j!},{\alpha }_{0,w}\ne 0.$We should note that the ${\mathrm{\Delta }}_w$-Gould–Hopper polynomials which are defined by the generating function [24] ${(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}=\sum _{k=0}^{\mathrm{\infty }}{G}_k^w(x,y)\frac{z^k}{k!},$form a bases for the ${\mathrm{\Delta }}_w$-Gould–Hopper polynomials and they are explicitly given by $G_k^w(x,y)=\sum _{m=0}^{\left[\frac{k}{2}\right]}(\genfrac{}{}{0ex}{}{k}{2m}){\left(x\right)}_{k-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}.$Motivated by the application areas of Appell polynomials and their above variants, in the present paper, we introduce the TI ${\mathrm{\Delta }}_w$-Gould–Hopper Appell polynomials and investigate some of their characteristic properties, which will have potential applications in different areas of applied sciences.

The paper is organized as follows: In Section 2, based on the definition of ${\mathrm{\Delta }}_w$-GHAP and discrete Appell convolutions, TI-DGHAP are introduced. Also, for this family of polynomials, the equivalence theorem, determinantal representation and circular theorem are obtained. In Section 3, we obtain a recurrence relation, LO, RO, DE, IPLO, IPRO and IPDE satisfied by the sequence of TI-DGHAP. In Section 4, we introduce TI degenerate Gould–Hopper Bernoulli polynomials, TI degenerate Gould–Hopper Poisson–Chalier polynomials, TI degenerate Gould–Hopper Boole polynomials and TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials for special cases of TI-DGHAP.

2. Some properties of the sequences of TI-DGHAP

In this section, we have examined the sequence of TI-DGHAP, which we have denote it with ${\mathcal{A}}_k^{\left[2\right]}(x,y;w):={\mathcal{A}}_k^{\left[2\right]}$. We present equivalance theorem, determinantal representation and circular theorem for this family of polynomials.

Let $Q_k(x,y)$ be the ${\mathrm{\Delta }}_w$-GHAP sequence given by (4) $Q_k(x,y)=\sum _{j=0}^k\sum _{m=0}^{\left[\frac{j}{2}\right]}(\genfrac{}{}{0ex}{}{k}{j})(\genfrac{}{}{0ex}{}{j}{2m})q_{k-j}{\left(x\right)}_{j-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}$(4) with (5) $a_1\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}{q}_k\frac{z^k}{k!},q_0\ne 0.$(5) Now we introduce the sequences of TI-DGHAP as follows:

Definition 2.1

Let $Q_j(x,y)$ be a sequence of ${\mathrm{\Delta }}_w$-GHAP given in (4(4) $Q_k(x,y)=\sum _{j=0}^k\sum _{m=0}^{\left[\frac{j}{2}\right]}(\genfrac{}{}{0ex}{}{k}{j})(\genfrac{}{}{0ex}{}{j}{2m})q_{k-j}{\left(x\right)}_{j-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}$(4) ) and $P_k(x,y)$ be any polynomial explicitly given as follows: (6) $P_k(x,y)=\sum _{j=0}^k\sum _{m=0}^{\left[\frac{j}{2}\right]}a(k,j)(\genfrac{}{}{0ex}{}{j}{2m}){\left(x\right)}_{j-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}.$(6) Let ${\mathcal{A}}_k^{\left[2\right]}$ be the polynomial convolution defined by the TI-DGHAP as (7) ${\mathcal{A}}_k^{\left[2\right]}:=\sum _{j=0}^{k}a(k,j)Q_j(x,y),k\in {\mathbb{N}}_0.$(7) The polynomials ${\mathcal{A}}_k^{\left[2\right]}$ will be called TI-DGHAP, if they satisfy the relation (8) ${}_x{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}=kw{\mathcal{A}}_{k-1}^{\left(2\right)},k\in \mathbb{N}.$(8)

Theorem 2.1

The following statements are equivalent.

(i)	$\{{\mathcal{A}}_k^{\left[2\right]}{\}}_{k\in \mathbb{N}}$ is a sequence of TI-DGHAP.
(ii)	$\left\{{\mathcal{P}}_k\right(x,y){\}}_{k\in \mathbb{N}}$ is a sequence of ${\mathrm{\Delta }}_w$-GHAP that satisfy given by $a(k,j)=(\genfrac{}{}{0ex}{}{k}{j})a_{k-j}$and $a_2\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}{a}_k\frac{z^k}{k!}.$
(iii)	The explicit representation of the sequence of TI-DGHAP $\{{\mathcal{A}}_k^{\left[2\right]}{\}}_{k\in \mathbb{N}}$ is as follows: (9) $\begin{array}{rl}{\mathcal{A}}_k^{\left[2\right]}& =\sum _{j=0}^k\sum _{l=0}^j\sum _{m=0}^{\left[\frac{l}{2}\right]}[(\genfrac{}{}{0ex}{}{k}{j})(\genfrac{}{}{0ex}{}{j}{l})(\genfrac{}{}{0ex}{}{l}{2m})a_{k-j}{q}_{j-l}\\ & \times {\left(x\right)}_{l-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}].\end{array}$(9)
(iv)	The generating function of the $\{{\mathcal{A}}_k^{\left[2\right]}{\}}_{k\in \mathbb{N}}$ is given by (10) $a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}.$(10)

Proof.

$\left(\mathit{\text{i}}\right)\iff \left(\mathit{\text{ii}}\right)$ Let $\{{\mathcal{A}}_k^{\left[2\right]}{\}}_{k\in \mathbb{N}}$ be a sequence of TI-DGHAP. Applying the finite difference operator ${}_x{\mathrm{\Delta }}_w$ on both sides of (7(7) ${\mathcal{A}}_k^{\left[2\right]}:=\sum _{j=0}^{k}a(k,j)Q_j(x,y),k\in {\mathbb{N}}_0.$(7) ), we get $\begin{array}{rl}kw{\mathcal{A}}_{k-1}^{\left[2\right]}& =\sum _{j=1}^{k}a(k,j)jw{Q}_{j-1}(x,y)\\ & =\sum _{j=0}^{k-1}a(k,j+1)(j+1)w{Q}_j(x,y).\end{array}$Therefore, we have ${\mathcal{A}}_{k-1}^{\left[2\right]}=\frac{1}{k}\sum _{j=0}^{k-1}a(k,j+1)(j+1)Q_j(x,y),$or equivalently (11) ${\mathcal{A}}_k^{\left[2\right]}=\frac{1}{k+1}\sum _{j=0}^{k}a(k+1,j+1)(j+1)Q_j(x,y).$(11) Comparing (7(7) ${\mathcal{A}}_k^{\left[2\right]}:=\sum _{j=0}^{k}a(k,j)Q_j(x,y),k\in {\mathbb{N}}_0.$(7) ) and (11(11) ${\mathcal{A}}_k^{\left[2\right]}=\frac{1}{k+1}\sum _{j=0}^{k}a(k+1,j+1)(j+1)Q_j(x,y).$(11) ), we have $a(k,j)=\frac{j+1}{k+1}a(k+1,j+1).$Iterating this last equation j times, we obtain $\begin{array}{rl}& a(k+1,j+1)\\ & =\frac{(k+1)k\cdots (k-j+1)}{(j+1)j\dots 1}a(k-j,0)\\ & =\frac{(k+1)k\cdots (k-j+1)(k-j)!}{(j+1)j\cdots 1(k-j)!}a(k-j,0)\\ & =(\genfrac{}{}{0ex}{}{k+1}{j+1})a(k-j,0).\end{array}$Hence we have (12) $a(k,j)=(\genfrac{}{}{0ex}{}{k}{j})a_{k-j}.$(12) Substituting (12(12) $a(k,j)=(\genfrac{}{}{0ex}{}{k}{j})a_{k-j}.$(12) ) in (6(6) $P_k(x,y)=\sum _{j=0}^k\sum _{m=0}^{\left[\frac{j}{2}\right]}a(k,j)(\genfrac{}{}{0ex}{}{j}{2m}){\left(x\right)}_{j-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}.$(6) ), we find that ${\mathcal{P}}_k(x,y)=\sum _{j=0}^k\sum _{m=0}^{\left[\frac{j}{2}\right]}{a}_{k-j}(\genfrac{}{}{0ex}{}{k}{j})(\genfrac{}{}{0ex}{}{j}{2m}){\left(x\right)}_{j-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}.$Using the finite difference operator ${}_x{\mathrm{\Delta }}_w$ on both sides of the last equation, we have ${}_x{\mathrm{\Delta }}_w\left({\mathcal{P}}_k(x,y)\right)=kw{\mathcal{P}}_{k-1}(x,y).$Therefore, $\left\{{\mathcal{P}}_k\right(x,y){\}}_{k\in \mathbb{N}}$ is a sequence of ${\mathrm{\Delta }}_w$-GHAP. The converse statement $\left(\mathit{\text{ii}}\right)\iff \left(\mathit{\text{i}}\right)$ is clearly demonstrated to be correct.

To obtain $\left(\mathit{\text{ii}}\right)\iff \left(\mathit{\text{iii}}\right)$, let $\left\{{\mathcal{P}}_k\right(x,y){\}}_{k\in \mathbb{N}}$ be a sequence of ${\mathrm{\Delta }}_w$-GHAP. By the definition of the TI-DGHAP, we have ${\mathcal{A}}_k^{\left[2\right]}:=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})a_{k-j}{Q}_j(x,y),k\in {\mathbb{N}}_0$where $Q_j(x,y)=\sum _{l=0}^j\sum _{m=0}^{\left[\frac{l}{2}\right]}(\genfrac{}{}{0ex}{}{j}{l})(\genfrac{}{}{0ex}{}{l}{2m})q_{j-l}{\left(x\right)}_{l-2m}^w{\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}.$Hence, the explicit form for TI-DGHAP is as follows: $\begin{array}{rl}{\mathcal{A}}_k^{\left[2\right]}& =\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})a_{k-j}\sum _{l=0}^j\sum _{m=0}^{\left[\frac{l}{2}\right]}[(\genfrac{}{}{0ex}{}{j}{l})(\genfrac{}{}{0ex}{}{l}{2m})q_{j-l}{\left(x\right)}_{l-2m}^w\\ & \times {\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}]\\ & =\sum _{j=0}^k\sum _{l=0}^j\sum _{m=0}^{\left[\frac{l}{2}\right]}[(\genfrac{}{}{0ex}{}{k}{j})(\genfrac{}{}{0ex}{}{j}{l})(\genfrac{}{}{0ex}{}{l}{2m})a_{k-j}{q}_{j-l}{\left(x\right)}_{l-2m}^w\\ & \times {\left(y\right)}_m^w\frac{\left(2m\right)!}{m!}].\end{array}$The converse proposition $\left(\mathit{\text{iii}}\right)\iff \left(\mathit{\text{ii}}\right)$ is clearly demonstrated to be correct. Lastly, to obtain $\left(\mathit{\text{iii}}\right)\iff \left(\mathit{\text{iv}}\right)$, writing the explicit form of ${\mathcal{A}}_k^{\left[2\right]}$ and applying the Cauchy product for the series, we get $\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}=a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}.$Thus, the proof is completed.

Theorem 2.2

The sequence of TI-DGHAP has the following determinantal representation: (13) $\begin{array}{rl}{A}_k^{\left[2\right]}& =\frac{{(-1)}^k}{{\left({\delta }_0\right)}^{k+1}}\\ & \times \left|\begin{array}{cccccc}{Q}_0(x,y)& Q_1(x,y)& & \cdots & Q_{k-1}(x,y)& Q_k(x,y)\\ {\delta }_0& {\delta }_1& & \cdots & {\delta }_{k-1}& {\delta }_k\\ 0& {\delta }_0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{1}){\delta }_{k-2}& (\genfrac{}{}{0ex}{}{k}{1}){\delta }_{k-1}\\ 0& 0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{2}){\delta }_{k-3}& (\genfrac{}{}{0ex}{}{k}{2}){\delta }_{k-2}\\ ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& & \cdots & {\delta }_0& (\genfrac{}{}{0ex}{}{k}{k-1}){\delta }_1\end{array}\right|\end{array}$(13) where ${\delta }_0,{\delta }_1,{\delta }_2,\dots ,{\delta }_n$ are the coefficients of the Maclaurin series of the function $\frac{1}{a_2\left(z\right)}$.

Proof.

Using the series representation of $\frac{1}{a_2\left(z\right)}$ as follows: ${\left[a_2\left(z\right)\right]}^{-1}=\sum _{j=0}^{\mathrm{\infty }}{\delta }_j\frac{z^j}{j!},$applying the generating function (10(10) $a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}.$(10) ), we get $\begin{array}{rl}& a_1\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & =\left(\sum _{j=0}^{\mathrm{\infty }}{\delta }_j\frac{z^j}{j!}\right)\left(\sum _{k=0}^{\mathrm{\infty }}{A}_k^{\left[2\right]}\frac{z^k}{k!}\right).\end{array}$Hence $\sum _{k=0}^{\mathrm{\infty }}{Q}_k(x,y)\frac{z^k}{k!}=\left(\sum _{j=0}^{\mathrm{\infty }}{\delta }_j\frac{z^j}{j!}\right)\left(\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{t^k}{k!}\right).$Applying the Cauchy product, we have $\sum _{k=0}^{\mathrm{\infty }}{Q}_k(x,y)\frac{z^k}{k!}=\sum _{k=0}^{\mathrm{\infty }}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\delta }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\frac{z^k}{k!}.$By comparing the coefficients of $\frac{z^k}{k!}$ from the polynomial equation, we get $Q_k(x,y)=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\delta }_j{\mathcal{A}}_{k-j}^{\left[2\right]},k\in {\mathbb{N}}_0.$As a result, we have the system of equations: $\begin{array}{rl}& Q_0(x,y)={\delta }_0{\mathcal{A}}_0^{\left[2\right]},\\ & Q_1(x,y)={\delta }_0{\mathcal{A}}_1^{\left[2\right]}+{\delta }_1{\mathcal{A}}_0^{\left[2\right]},\\ & Q_2(x,y)={\delta }_0{\mathcal{A}}_2^{\left[2\right]}+(\genfrac{}{}{0ex}{}{2}{1}){\delta }_1{\mathcal{A}}_1^{\left[2\right]}+{\delta }_2{\mathcal{A}}_0^{\left[2\right]},\\ & ⋮\\ & Q_{k-1}(x,y)={\delta }_0{\mathcal{A}}_{k-1}^{\left[2\right]}+(\genfrac{}{}{0ex}{}{k-1}{1}){\delta }_1{\mathcal{A}}_{k-2}^{\left[2\right]}+\dots {\delta }_{k-1}{\mathcal{A}}_0^{\left[2\right]},\\ & Q_k(x,y)={\delta }_0{\mathcal{A}}_k^{\left[2\right]}+(\genfrac{}{}{0ex}{}{k}{1}){\delta }_1{\mathcal{A}}_{k-1}^{\left[2\right]}+\dots +{\delta }_k{\mathcal{A}}_0^{\left[2\right]}.\end{array}$Applying Cramer's rule, we get $A_k^{\left[2\right]}=\frac{\left|\begin{array}{cccccc}{\delta }_0& 0& & \cdots & 0& Q_0(x,y)\\ {\delta }_1& {\delta }_0& & \cdots & 0& Q_1(x,y)\\ {\delta }_2& (\genfrac{}{}{0ex}{}{2}{1}){\delta }_1& & \cdots & 0& Q_2(x,y)\\ {\delta }_3& (\genfrac{}{}{0ex}{}{3}{2}){\delta }_2& & \cdots & 0& Q_3(x,y)\\ ⋮& ⋮& & ⋮& ⋮& ⋮\\ {\delta }_{k-1}& (\genfrac{}{}{0ex}{}{k-1}{1}){\delta }_{k-2}& & \cdots & {\delta }_0& Q_{k-1}(x,y)\\ {\delta }_k& (\genfrac{}{}{0ex}{}{k}{1}){\delta }_{k-1}& & \cdots & (\genfrac{}{}{0ex}{}{k}{k-1}){\delta }_1& Q_k(x,y)\end{array}\right|}{\left|\begin{array}{cccccc}{\delta }_0& 0& & \cdots & 0& 0\\ {\delta }_1& {\delta }_0& & \cdots & 0& 0\\ {\delta }_2& (\genfrac{}{}{0ex}{}{2}{1}){\delta }_1& & \cdots & 0& 0\\ {\delta }_3& (\genfrac{}{}{0ex}{}{3}{2}){\delta }_2& & \cdots & 0& 0\\ ⋮& ⋮& & ⋮& ⋮& ⋮\\ {\delta }_{k-1}& (\genfrac{}{}{0ex}{}{k-1}{1}){\delta }_{k-2}& & \cdots & {\delta }_0& 0\\ {\delta }_k& (\genfrac{}{}{0ex}{}{k}{1}){\delta }_{k-1}& & \cdots & (\genfrac{}{}{0ex}{}{k}{k-1}){\delta }_1& {\delta }_0\end{array}\right|}$By taking the transpose in the last equation, we have $\begin{array}{rl}{A}_k^{\left[2\right]}& =\frac{1}{{\left({\delta }_0\right)}^{k+1}}\\ & \times \left|\begin{array}{cccccc}{\delta }_0& {\delta }_1& & \cdots & {\delta }_{k-1}& {\delta }_k\\ 0& {\delta }_0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{1}){\delta }_{k-2}& (\genfrac{}{}{0ex}{}{k}{1}){\delta }_{k-1}\\ 0& 0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{2}){\delta }_{k-3}& (\genfrac{}{}{0ex}{}{k}{2}){\delta }_{k-2}\\ 0& 0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{3}){\delta }_{k-4}& (\genfrac{}{}{0ex}{}{k}{3}){\delta }_{k-3}\\ ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& & \cdots & {\delta }_0& (\genfrac{}{}{0ex}{}{k}{k-1}){\delta }_1\\ Q_0(x,y)& Q_1(x,y)& & \cdots & Q_{k-1}(x,y)& Q_k(x,y)\end{array}\right|.\end{array}$As a result the proof is completed using the elementary row operations.

Theorem 2.3

The sequence of polynomials having the form (13) satisfies (14) ${}_x{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}=kw{\mathcal{A}}_{k-1}^{\left[2\right]},k\in \mathbb{N}.$(14)

Proof.

Applying ${}_x{\mathrm{\Delta }}_w$ to the determinantal representation given in (13), we obtain

$\begin{array}{rl}& {}_x{\mathrm{\Delta }}_w{A}_k^{\left[2\right]}=\frac{{(-1)}^k}{{\left({\delta }_0\right)}^{k+1}}\left|\begin{array}{cccccc}{}_x{\mathrm{\Delta }}_w{Q}_0(x,y)& {}_x{\mathrm{\Delta }}_w{Q}_1(x,y)& & \cdots & {}_x{\mathrm{\Delta }}_w{Q}_{k-1}(x,y)& {}_x{\mathrm{\Delta }}_w{Q}_k(x,y)\\ {\delta }_0& {\delta }_1& & \cdots & {\delta }_{k-1}& {\delta }_k\\ 0& {\delta }_0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{1}){\delta }_{k-2}& (\genfrac{}{}{0ex}{}{k}{1}){\delta }_{k-1}\\ 0& 0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{2}){\delta }_{k-3}& (\genfrac{}{}{0ex}{}{k}{2}){\delta }_{k-2}\\ ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& & \cdots & {\delta }_0& (\genfrac{}{}{0ex}{}{k}{k-1}){\delta }_1\end{array}\right|.\end{array}$

Using the relation ${}_x{\mathrm{\Delta }}_w{Q}_k(x,y)=kw{Q}_{k-1}(x,y)$and then dividing the first row by w, we can expand the determinant with respect to the first column. The proof is completed by multiplying the lth row by l−1, and the mth column by $\frac{1}{m}$.

Theorem 2.4

Circular Theorem

The following statements are equivalent for the sequence of TI-DGHAP:

(i)	$\{{\mathcal{A}}_k^{\left[2\right]}{\}}_{k\in \mathbb{N}}$ is a TI-DGHAP.
(ii)	$\left(\mathit{\text{ii}}\right)$ The TI-DGHAP ${\mathcal{A}}_k^{\left[2\right]}$ has the generating function given by $a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}.$
(iii)	The TI-DGHAP ${\mathcal{A}}_k^{\left[2\right]}$ are expressed in a determinantal representation in (13(13) $\begin{array}{rl}{A}_k^{\left[2\right]}& =\frac{{(-1)}^k}{{\left({\delta }_0\right)}^{k+1}}\\ & \times \left|\begin{array}{cccccc}{Q}_0(x,y)& Q_1(x,y)& & \cdots & Q_{k-1}(x,y)& Q_k(x,y)\\ {\delta }_0& {\delta }_1& & \cdots & {\delta }_{k-1}& {\delta }_k\\ 0& {\delta }_0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{1}){\delta }_{k-2}& (\genfrac{}{}{0ex}{}{k}{1}){\delta }_{k-1}\\ 0& 0& & \cdots & (\genfrac{}{}{0ex}{}{k-1}{2}){\delta }_{k-3}& (\genfrac{}{}{0ex}{}{k}{2}){\delta }_{k-2}\\ ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& & \cdots & {\delta }_0& (\genfrac{}{}{0ex}{}{k}{k-1}){\delta }_1\end{array}\right|\end{array}$(13) ).

Proof.

The proof of this theorem is clearly seen using Theorems 2.1, 2.2 and 2.3.

3. Reccurence relation, LO, RO, DE, IPLO, IPRO and IPDE of the TI-DGHAP

In this section, we obtain the recurrence relation, LO, RO, DE, IPLO, IPRO and IPDE which are satisfied by the TI-DGHAP.

Theorem 3.1

A recurrence relation satisfied by TI-DGHAP is given by (15) $\begin{array}{rl}{\mathcal{A}}_{k+1}^{\left[2\right]}& =\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\theta }_j{\mathcal{A}}_{k-j}^{\left[2\right]}+\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\rho }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\\ & +x\sum _{j=0}^k\frac{k!}{(k-j)!}{(-w)}^j{\mathcal{A}}_{k-j}^{\left[2\right]}\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{k!}{(k-2j-1)!}{(-w)}^j{\mathcal{A}}_{k-2j-1}^{\left[2\right]},k\ge 1\end{array}$(15) where (16) $\frac{a_1^{\mathrm{\prime }}\left(z\right)}{a_1\left(z\right)}=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j\frac{z^j}{j!}\mathrm{a}\mathrm{n}\mathrm{d}\frac{a_2^{\mathrm{\prime }}\left(z\right)}{a_2\left(z\right)}=\sum _{j=0}^{\mathrm{\infty }}{\rho }_j\frac{z^j}{j!}.$(16)

Proof.

If we take the derivative with respect to t on both sides of (10(10) $a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}.$(10) ), we get $\begin{array}{rl}& \sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_{k+1}^{\left[2\right]}\frac{z^k}{k!}\\ & =\frac{a_1^{\mathrm{\prime }}\left(z\right)}{a_1\left(z\right)}{a}_1\left(z\right)a_2\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & +\frac{a_2^{\mathrm{\prime }}\left(z\right)}{a_2\left(z\right)}{a}_1\left(z\right)a_2\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & +\frac{x}{1+wz}{a}_1\left(z\right)a_2\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & +\frac{2yz}{1+w{z}^2}{a}_1\left(z\right)a_2\left(z\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}.\end{array}$Using (16(16) $\frac{a_1^{\mathrm{\prime }}\left(z\right)}{a_1\left(z\right)}=\sum _{j=0}^{\mathrm{\infty }}{\theta }_j\frac{z^j}{j!}\mathrm{a}\mathrm{n}\mathrm{d}\frac{a_2^{\mathrm{\prime }}\left(z\right)}{a_2\left(z\right)}=\sum _{j=0}^{\mathrm{\infty }}{\rho }_j\frac{z^j}{j!}.$(16) ) and the series expansions: (17) $\begin{array}{rl}\frac{1}{1+wz}& =\sum _{j=0}^{\mathrm{\infty }}{(-wz)}^j,|-wz|<1;\\ & \frac{1}{1+w{z}^2}=\sum _{j=0}^{\mathrm{\infty }}{(-w{z}^2)}^j,|-w{z}^2|<1,\end{array}$(17) we have $\begin{array}{rl}\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_{k+1}^{\left[2\right]}\frac{z^k}{k!}& =\left(\sum _{j=0}^{\mathrm{\infty }}{\theta }_j\frac{z^j}{j!}\right)\left(\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}\right)\\ & +\left(\sum _{k=0}^{\mathrm{\infty }}{\rho }_j\frac{z^j}{j!}\right)\left(\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}\right)\\ & +x\left(\sum _{j=0}^{\mathrm{\infty }}{(-wz)}^j\right)\left(\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^k}{k!}\right)\\ & +2y\left(\sum _{j=0}^{\mathrm{\infty }}{(-w{z}^2)}^j\right)\left(\sum _{n=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[2\right]}\frac{z^{k+1}}{k!}\right).\end{array}$Hence, by using the Cauchy product, we obtain (18) $\begin{array}{rl}& \sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_{k+1}^{\left[2\right]}\frac{z^k}{k!}\\ & =(x+{\theta }_0+{\rho }_0){\mathcal{A}}_0^{\left[2\right]}+\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\theta }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\frac{z^k}{k!}\\ & +\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\rho }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\frac{z^k}{k!}\\ & +x\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^k\frac{k!}{(k-j)!}{(-w)}^j{\mathcal{A}}_{k-j}^{\left[2\right]}\frac{z^k}{k!}\\ & +2y\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^{\left[\frac{k-1}{2}\right]}[\frac{k!}{(k-2j-1)!}{(-w)}^j{\mathcal{A}}_{k-2j-1}^{\left[2\right]}\\ & \times \frac{z^k}{k!}],k\ge 1.\end{array}$(18) The proof is completed by equating the coefficients of $\frac{z^k}{k!}$ on both sides of the (18(18) $\begin{array}{rl}& \sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_{k+1}^{\left[2\right]}\frac{z^k}{k!}\\ & =(x+{\theta }_0+{\rho }_0){\mathcal{A}}_0^{\left[2\right]}+\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\theta }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\frac{z^k}{k!}\\ & +\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\rho }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\frac{z^k}{k!}\\ & +x\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^k\frac{k!}{(k-j)!}{(-w)}^j{\mathcal{A}}_{k-j}^{\left[2\right]}\frac{z^k}{k!}\\ & +2y\sum _{k=1}^{\mathrm{\infty }}\sum _{j=0}^{\left[\frac{k-1}{2}\right]}[\frac{k!}{(k-2j-1)!}{(-w)}^j{\mathcal{A}}_{k-2j-1}^{\left[2\right]}\\ & \times \frac{z^k}{k!}],k\ge 1.\end{array}$(18) ).

Theorem 3.2

The LO, RO and DE satisfied by the TI-DGHAP are given by (19) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{kw}{}_x{\mathrm{\Delta }}_w,\end{array}$(19) (20) $\begin{array}{rl}{L}_k^{+}:& =\sum _{j=0}^k\frac{{\theta }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}[\\ & \times \frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1}],k\ge 1\end{array}$(20) and (21) $\begin{array}{rl}& [\sum _{j=0}^k\frac{{\theta }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^{j+1}+\sum _{j=0}^k\frac{{\rho }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +(x+w)\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}\\ & -(k+1)w\phantom{+w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}}]{\mathcal{A}}_k^{\left[2\right]}=0,k\ge 1\end{array}$(21) respectively.

Proof.

Using the following relation: ${}_x{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}=kw{\mathcal{A}}_{k-1}^{\left[2\right]},$we get $\frac{1}{kw}{}_x{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}={\mathcal{A}}_{k-1}^{\left[2\right]},$which clearly shows that the lowering operator is given by $L_k^{-}:=\frac{1}{kw}{}_x{\mathrm{\Delta }}_w.$We can write the term ${\mathcal{A}}_{k-j}^{\left[2\right]}$ by applying the lowering operator to the term ${\mathcal{A}}_k^{\left[2\right]}$ j times as follows: (22) $\begin{array}{rl}{\mathcal{A}}_{k-j}^{\left[2\right]}& =[L_{k-j+1}^{-}{L}_{k-j+2}^{-}\dots L_k^{-}]{\mathcal{A}}_k^{\left[2\right]}\\ & =[\frac{1}{(k-j+1)w}{}_x{\mathrm{\Delta }}_w\frac{1}{(k-j+2)w}{}_x{\mathrm{\Delta }}_w\dots \frac{1}{kw}{}_x{\mathrm{\Delta }}_w]{\mathcal{A}}_k^{\left[2\right]}\\ & =\frac{(k-j)!}{k!w^j}{}_x{\mathrm{\Delta }}_w^j{\mathcal{A}}_k^{\left[2\right]}.\end{array}$(22)

Similarly, we obtain ${\mathcal{A}}_{k-2j-1}^{\left[2\right]}$ as follows: (23) ${\mathcal{A}}_{k-2j-1}^{\left[2\right]}=\frac{(k-2j-1)!}{k!w^{2j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1}{\mathcal{A}}_k^{\left[2\right]}.$(23) Upon substituting from (22(22) $\begin{array}{rl}{\mathcal{A}}_{k-j}^{\left[2\right]}& =[L_{k-j+1}^{-}{L}_{k-j+2}^{-}\dots L_k^{-}]{\mathcal{A}}_k^{\left[2\right]}\\ & =[\frac{1}{(k-j+1)w}{}_x{\mathrm{\Delta }}_w\frac{1}{(k-j+2)w}{}_x{\mathrm{\Delta }}_w\dots \frac{1}{kw}{}_x{\mathrm{\Delta }}_w]{\mathcal{A}}_k^{\left[2\right]}\\ & =\frac{(k-j)!}{k!w^j}{}_x{\mathrm{\Delta }}_w^j{\mathcal{A}}_k^{\left[2\right]}.\end{array}$(22) ) and (23(23) ${\mathcal{A}}_{k-2j-1}^{\left[2\right]}=\frac{(k-2j-1)!}{k!w^{2j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1}{\mathcal{A}}_k^{\left[2\right]}.$(23) ) in (15(15) $\begin{array}{rl}{\mathcal{A}}_{k+1}^{\left[2\right]}& =\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\theta }_j{\mathcal{A}}_{k-j}^{\left[2\right]}+\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\rho }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\\ & +x\sum _{j=0}^k\frac{k!}{(k-j)!}{(-w)}^j{\mathcal{A}}_{k-j}^{\left[2\right]}\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{k!}{(k-2j-1)!}{(-w)}^j{\mathcal{A}}_{k-2j-1}^{\left[2\right]},k\ge 1\end{array}$(15) ), we get $\begin{array}{rl}{\mathcal{A}}_{k+1}^{\left[2\right]}& =[\sum _{j=0}^k\frac{{\theta }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1}]\\ & \times {\mathcal{A}}_k^{\left[2\right]},k\ge 1.\end{array}$Hence the raising operator is given by $\begin{array}{rl}{L}_k^{+}& :=\sum _{j=0}^k\frac{{\theta }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1},\\ & k\ge 1.\end{array}$Now, using the factorization method to obtain the difference equation of the TI-DGHAP $L_{k+1}^{-}{L}_k^{+}\left({\mathcal{A}}_k^{\left[2\right]}\right)={\mathcal{A}}_k^{\left[2\right]},$we get $\begin{array}{rl}{\mathcal{A}}_k^{\left[2\right]}& =\frac{1}{(k+1)w}{}_x{\mathrm{\Delta }}_w[\sum _{j=0}^k\frac{{\theta }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1}]{\mathcal{A}}_k^{\left[2\right]},k\ge 1.\end{array}$In view of the following the relation in [15] $\begin{array}{rl}& {}_x{\mathrm{\Delta }}_w\left\{u\left(x\right)v\left(x\right)\right\}\\ & =u(x+w){}_x{\mathrm{\Delta }}_w\left\{v\left(x\right)\right\}+v\left(x\right){}_x{\mathrm{\Delta }}_w\left\{u\left(x\right)\right\}\end{array}$and when the terms are reorganized, we can write $\begin{array}{rl}& [\sum _{j=0}^k\frac{{\theta }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^{j+1}+\sum _{j=0}^k\frac{{\rho }_j}{j!}{w}^{-j}{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +(x+w)\sum _{j=0}^k{(-1)}_x^j{\mathrm{\Delta }}_w^{j+1}\\ & +w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}\\ & -\phantom{w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}}(k+1)w]{\mathcal{A}}_k^{\left[2\right]}=0,k\ge 1.\end{array}$Thus the proof is completed.

Theorem 3.3

The IPLO, IPRO and IPDE satisfied by the TI-DGHAP are given by (24) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{k}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w,\end{array}$(24) (25) $\begin{array}{rl}{L}_k^{+}:& =\sum _{j=0}^k\frac{{\theta }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1},k\ge 1\end{array}$(25) and (26) $\begin{array}{rl}& [\sum _{j=0}^k\frac{{\theta }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}+\sum _{j=0}^k\frac{{\rho }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +x\sum _{j=0}^n{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +2(y+w)\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+2}\\ & +2w\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}-(k+1){}_x{\mathrm{\Delta }}_w]\\ & \times {\mathcal{A}}_k^{\left[2\right]}=0,k\ge 1\end{array}$(26) where ${}_x{\mathrm{\Delta }}_w^{-1}$ is the inverse of ${}_x{\mathrm{\Delta }}_w$, respectively.

Proof.

Using the following relations: ${}_x{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}=kw{\mathcal{A}}_{k-1}^{\left[2\right]}$and ${}_y{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}=k(k-1)w{\mathcal{A}}_{k-2}^{\left[2\right]},$we get ${}_y{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}=k{}_x{\mathrm{\Delta }}_w{\mathcal{A}}_{k-1}^{\left[2\right]}.$By applying ${}_x{\mathrm{\Delta }}_w^{-1}$ to both sides of the last equation, we have ${\mathcal{A}}_{k-1}^{\left[2\right]}=\frac{1}{k}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]},$which states that the integro-partial lowering operator is given by $L_k^{-}:=\frac{1}{k}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w.$Similarly, using (8(8) ${}_x{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}=kw{\mathcal{A}}_{k-1}^{\left(2\right)},k\in \mathbb{N}.$(8) ), we have $\frac{1}{kw}{}_x{\mathrm{\Delta }}_w{\mathcal{A}}_k^{\left[2\right]}={\mathcal{A}}_{k-1}^{\left[2\right]}.$By applying ${}_x{\mathrm{\Delta }}_w^{-1}$ to both sides of the equation and using arithmetic operations, we get ${}_x{\mathrm{\Delta }}_w^{-j}{\mathcal{A}}_{k-1}^{\left[2\right]}=\frac{1}{k(k+1)\dots (k+j-1)w^j}{\mathcal{A}}_{k+j-1}^{\left[2\right]}.$By applying the IPLO, k−j times, to the term ${\mathcal{A}}_k^{\left[2\right]}$, we have $\begin{array}{rl}{\mathcal{A}}_j^{\left[2\right]}& =[{\mathcal{L}}_{j+1}^{-}{\mathcal{L}}_{j+2}^{-}\dots {\mathcal{L}}_k^{-}]{\mathcal{A}}_k^{\left[2\right]}\\ & =[\frac{1}{(j+1)}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w\frac{1}{(j+2)}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w\dots \frac{1}{k}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w]{\mathcal{A}}_k^{\left[2\right]}\\ & =\frac{j!}{k!}{}_x{\mathrm{\Delta }}_w^{-(k-j)}{}_y{\mathrm{\Delta }}_w^{k-j}{\mathcal{A}}_k^{\left[2\right]}.\end{array}$

Similarly, we obtain ${\mathcal{A}}_{k-j}^{\left[2\right]}$ and ${\mathcal{A}}_{k-2j-1}^{\left[2\right]}$ as follows: (27) ${\mathcal{A}}_{k-j}^{\left[2\right]}=\frac{(k-j)!}{k!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j{\mathcal{A}}_k^{\left[2\right]}$(27) and (28) ${\mathcal{A}}_{k-2j-1}^{\left[2\right]}=\frac{(k-2j-1)!}{k!}{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}{\mathcal{A}}_k^{\left[2\right]}.$(28) By using the (27(27) ${\mathcal{A}}_{k-j}^{\left[2\right]}=\frac{(k-j)!}{k!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j{\mathcal{A}}_k^{\left[2\right]}$(27) ), (28(28) ${\mathcal{A}}_{k-2j-1}^{\left[2\right]}=\frac{(k-2j-1)!}{k!}{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}{\mathcal{A}}_k^{\left[2\right]}.$(28) ) in (15(15) $\begin{array}{rl}{\mathcal{A}}_{k+1}^{\left[2\right]}& =\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\theta }_j{\mathcal{A}}_{k-j}^{\left[2\right]}+\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\rho }_j{\mathcal{A}}_{k-j}^{\left[2\right]}\\ & +x\sum _{j=0}^k\frac{k!}{(k-j)!}{(-w)}^j{\mathcal{A}}_{k-j}^{\left[2\right]}\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{k!}{(k-2j-1)!}{(-w)}^j{\mathcal{A}}_{k-2j-1}^{\left[2\right]},k\ge 1\end{array}$(15) ), we get $\begin{array}{rl}{\mathcal{A}}_{k+1}^{\left[2\right]}& =[\sum _{j=0}^k\frac{{\theta }_k}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}({(-w)}^j\\ & \times {}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1})]{\mathcal{A}}_k^{\left[2\right]},k\ge 1.\end{array}$Hence the integro-partial raising operator is given by $\begin{array}{rl}{L}_k^{+}:& =\sum _{j=0}^k\frac{{\theta }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1},k\ge 1.\end{array}$Now, using the factorization method to obtain the integro-partial difference equation of the TI-DGHAP ${\mathcal{L}}_{k+1}^{-}{\mathcal{L}}_k^{+}\left({\mathcal{A}}_k^{\left[2\right]}\right))={\mathcal{A}}_k^{\left[2\right]},$we get $\begin{array}{rl}& \frac{1}{k+1}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w[\sum _{j=0}^k\frac{{\theta }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\frac{{\rho }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}]{\mathcal{A}}_k^{\left[2\right]}\\ & ={\mathcal{A}}_k^{\left[2\right]},k\ge 1\end{array}$and reorganizing the terms, we can write $\begin{array}{rl}& [\sum _{j=0}^k\frac{{\theta }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}+\sum _{j=0}^k\frac{{\rho }_j}{j!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +2(y+w)\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+2}\\ & +2w\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}-(k+1){}_x{\mathrm{\Delta }}_w]\\ & \times {\mathcal{A}}_k^{\left[2\right]}=0,k\ge 1.\end{array}$Thus the proof is completed.

4. Special cases of the TI-DGHAP

In this section, some special cases of the determining functions $a_1\left(z\right)$ and $a_2\left(z\right)$, we define TI degenerate Gould–Hopper Bernoulli polynomials, TI degenerate Gould–Hopper Poisson–Chalier polynomials, TI degenerate Gould–Hopper Boole polynomials and TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials and obtain the recurrence relations, LO, RO, DE, IPLO, IPRO and IPDE for these polynomials.

4.1. TI degenerate Gould–Hopper Bernoulli polynomials

The TI degenerate Gould–Hopper Bernoulli polynomials ${\mathcal{B}}_k^{\left[2\right]}(x,y;w;\alpha ;\text{β}):={\mathcal{B}}_k^{\left[2\right]}$ are defined as follows: (29) $\begin{array}{rl}& [{\left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right)}^{\alpha }{\left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right)}^{\text{β}}{(1+wz)}^{\frac{x}{w}}\\ & \times {(1+w{z}^2)}^{\frac{y}{w}}]=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{B}}_k^{\left[2\right]}\frac{z^k}{k!}\end{array}$(29) where $\begin{array}{rl}{a}_1\left(z\right)& ={\left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right)}^{\alpha }\mathrm{a}\mathrm{n}\mathrm{d}\\ a_2\left(z\right)& ={\left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right)}^{\text{β}}.\end{array}$

Corollary 4.1

The recurrence relation satisfied by the TI degenerate Gould–Hopper Bernoulli polynomials is as follows: (30) $\begin{array}{rl}{\mathcal{B}}_{k+1}^{\left[2\right]}& =-(\alpha +\text{β})[\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){(-w)}^{j}j!{\mathcal{B}}_{k-j}^{\left[2\right]}\\ & +\sum _{j=0}^k\sum _{m=0}^{j+1}(\genfrac{}{}{0ex}{}{k}{j}){(-w)}^{j-m+1}j!\frac{{\mathcal{B}}_{m,w}}{m!}{\mathcal{B}}_{k-j}^{\left[2\right]}]\\ & +x\sum _{j=0}^k{(-w)}^j\frac{k!}{(k-j)!}{\mathcal{B}}_{k-j}^{\left[2\right]}\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j\frac{k!}{(k-2j-1)!}{\mathcal{B}}_{k-2j-1}^{\left[2\right]},k\ge 1.\end{array}$(30) where the degenerate Bernoulli numbers ${\mathcal{B}}_{m,w}$ are given by the following series in [25] (31) $\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{B}}_{m,w}\frac{z^m}{m!}.$(31)

Corollary 4.2

The LO, RO and DE satisfied by the TI degenerate Gould–Hopper Bernoulli polynomials are as follows: (32) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{kw}{}_x{\mathrm{\Delta }}_w,\end{array}$(32) (33) $\begin{array}{rl}{L}_k^{+}:& =-(\alpha +\text{β})[\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+\sum _{j=0}^k\sum _{m=0}^{j+1}(\\ & \times {(-1)}^{j-m+1}{w}^{-m+1}\frac{{\mathcal{B}}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^j)\phantom{\sum _{j=0}^k{(-1)}^j{}_x}]\\ & +x\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1}\end{array}$(33) and (34) $\begin{array}{rl}& [-(\alpha +\text{β})\sum _{j=0}^k\sum _{m=0}^{j+1}{(-1)}^{j-m+1}{w}^{-m+1}\frac{{\mathcal{B}}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{j+1}\\ & -(\alpha +\text{β})\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}+(x+w)\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}-(k+1)w]\\ & \times {\mathcal{B}}_k^{\left[2\right]}=0,k\ge 1\end{array}$(34)

where ${\mathcal{B}}_{m,w}$ are the degenerate Bernoulli numbers in (31(31) $\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{B}}_{m,w}\frac{z^m}{m!}.$(31) ), respectively.

Corollary 4.3

The IPLO, IPRO and IPDE satisfied by the TI degenerate Gould–Hopper Bernoulli polynomials are as follows: (35) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{k}{}_x{\mathrm{\Delta }}_w{}_y{\mathrm{\Delta }}_w^{-1},\end{array}$(35) (36) $\begin{array}{rl}{L}_k^{+}:& =-(\alpha +\text{β})[\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +\sum _{j=0}^n\sum _{m=0}^{j+1}{(-w)}^{j-m+1}\frac{{\mathcal{B}}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j]\\ & +x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}\end{array}$(36) and (37) $\begin{array}{rl}& [-(\alpha +\text{β})\sum _{j=0}^k\sum _{m=0}^{j+1}{(-w)}^{j-m+1}\frac{{\mathcal{B}}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & -(\alpha +\text{β})\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}+2(y+w)\\ & \times (\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+2})\\ & +2w\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}-(k+1){}_x{\mathrm{\Delta }}_w]\\ & \times {\mathcal{B}}_k^{\left[2\right]}=0,k\ge 1\end{array}$(37) where ${\mathcal{B}}_{m,w}$ are the degenerate Bernoulli numbers in (31(31) $\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}=\sum _{m=0}^{\mathrm{\infty }}{\mathcal{B}}_{m,w}\frac{z^m}{m!}.$(31) ), respectively.

Remark 4.1

When we write $\alpha =1$ and $\text{β}=0$ in the TI degenerate Gould–Hopper Bernoulli polynomials, we get degenerate Gould–Hopper polynomials for the following generating function in [15] $\begin{array}{rl}& \left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & =\sum _{k=0}^{\mathrm{\infty }}{\mathcal{B}}_k^{\left[2\right]}\frac{z^k}{k!}.\end{array}$When we write $\alpha =1$ and $\text{β}=1$ in the TI degenerate Gould–Hopper Bernoulli polynomials, we get degenerate Gould–Hopper Bernoulli polynomials of order 2 for the following generating function $\begin{array}{rl}& {\left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right)}^2{(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & =\sum _{k=0}^{\mathrm{\infty }}{\mathcal{B}}_k^{\left[2\right]}\frac{z^k}{k!}.\end{array}$

Remark 4.2

We should notice that if we try to find the recurrence relation and difference equation of degenerate Gould–Hopper Bernoulli polynomials of order 2, the coefficient will be given of in terms of the degenerate Gould–Hopper Bernoulli polynomials the number of order 2. But we are seen from Corollary 4.1 and 4.2 that the recurrence relation and difference equation is given in terms of the Bernoulli numbers of order 1 generating by (29(29) $\begin{array}{rl}& [{\left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right)}^{\alpha }{\left(\frac{z}{{(1+wz)}^{\frac{1}{w}}-1}\right)}^{\text{β}}{(1+wz)}^{\frac{x}{w}}\\ & \times {(1+w{z}^2)}^{\frac{y}{w}}]=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{B}}_k^{\left[2\right]}\frac{z^k}{k!}\end{array}$(29) ).

4.2. TI degenerate Gould–Hopper Poisson–Charlier polynomials

The TI degenerate Gould–Hopper Poisson–Charlier polynomials $C_k^{\left[2\right]}(x,y;w;\alpha ;\text{β}):=C_k^{\left[2\right]}$ are defined as follows: (38) ${\mathrm{e}}^{-{\alpha }^{w}z}{\mathrm{e}}^{-{\text{β}}^{w}z}{(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}=\sum _{n=0}^{\mathrm{\infty }}{C}_k^{\left[2\right]}\frac{z^k}{k!},$(38) where $a_1\left(z\right)={\mathrm{e}}^{-{\alpha }^{w}z}\mathrm{a}\mathrm{n}\mathrm{d}{a}_2\left(z\right)={\mathrm{e}}^{-{\text{β}}^{w}z}.$

Corollary 4.4

The recurrence relation satisfied by the TI degenerate Gould–Hopper Poisson–Charlier polynomials is as follows: (39) $\begin{array}{rl}{C}_{k+1}^{\left[2\right]}& =-({\alpha }^w+{\text{β}}^w)C_k^{\left[2\right]}+x\sum _{j=0}^k{(-w)}^j\frac{k!}{(k-j)!}{C}_{k-j}^{\left[2\right]}\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j\frac{k!}{(k-2j-1)!}{C}_{k-2j-1}^{\left[2\right]},k\ge 1.\end{array}$(39)

Corollary 4.5

The LO, RO and DE satisfied by the TI degenerate Gould–Hopper Poisson–Charlier polynomials are as follows: (40) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{kw}{}_x{\mathrm{\Delta }}_w,\end{array}$(40) (41) $\begin{array}{rl}{L}_k^{+}:& =-({\alpha }^w+{\text{β}}^w)+x\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1},k\ge 1\end{array}$(41) and (42) $\begin{array}{rl}& [-({\alpha }^w+{\text{β}}^w){}_x{\mathrm{\Delta }}_w+(x+w)\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}\\ & -(k+1)w]C_k^{\left[2\right]}=0,k\ge 1\end{array}$(42) respectively.

Corollary 4.6

The IPLO, IPRO and IPDE satisfied by the TI degenerate Gould–Hopper Poisson–Charlier polynomials are as follows: (43) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{k}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w,\end{array}$(43) (44) $\begin{array}{rl}{L}_k^{+}:& =-({\alpha }^w+{\text{β}}^w)+x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1},k\ge 1\end{array}$(44) and (45) $\begin{array}{rl}& [-({\alpha }^w+{\text{β}}^w){}_y{\mathrm{\Delta }}_w+x\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +2(y+w)\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+2}\\ & +2w\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}-(k+1){}_x{\mathrm{\Delta }}_w]\\ & \times C_k^{\left[2\right]}=0,k\ge 1\end{array}$(45) respectively.

4.3. TI degenerate Gould–Hopper Boole polynomials

The TI degenerate Gould–Hopper Boole polynomials $B{l}_k^{\left[2\right]}(x,y;w;\eta ):=B{l}_k^{\left[2\right]}$ are defined as follows: (46) $\begin{array}{rl}& {\left(\frac{1}{1+{(1+wz)}^{\frac{\eta }{w}}}\right)}^2{(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & =\sum _{k=0}^{\mathrm{\infty }}B{l}_k^{\left[2\right]}\frac{z^k}{k!}\end{array}$(46) where $a_1\left(z\right)=a_2\left(z\right)=\frac{1}{1+{(1+wz)}^{\frac{\eta }{w}}}.$

Corollary 4.7

The recurrence relations satisfied by the TI degenerate Gould–Hopper Boole polynomials is as follows: (47) $\begin{array}{rl}B{l}_{k+1}^{\left[2\right]}& =-2\eta \sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){(-w)}^{j}j!B{l}_{k-j}^{\left[2\right]}\\ & +2\eta \sum _{j=0}^k\sum _{m=0}^j(\genfrac{}{}{0ex}{}{k}{j}){(-w)}^{j-m}j!\frac{B{l}_{m,w}}{m!}B{l}_{k-j}^{\left[2\right]}\\ & +x\sum _{j=0}^k{(-w)}^j\frac{k!}{(k-j)!}B{l}_{k-j}^{\left[2\right]}\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j\frac{k!}{(k-2j-1)!}B{l}_{k-2j-1}^{\left[2\right]},k\ge 1\end{array}$(47) where $B{l}_{m,w}$ are the degenerate Boole numbers generating by in [24] (48) $\frac{1}{1+{(1+z)}^{\frac{\eta }{w}}}=\sum _{m=0}^{\mathrm{\infty }}B{l}_{m,w}\frac{z^m}{m!}.$(48)

Corollary 4.8

The LO, RO and DE satisfied by the TI degenerate Gould–Hopper Boole polynomials are as follows: (49) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{kw}{}_x{\mathrm{\Delta }}_w,\end{array}$(49) (50) $\begin{array}{rl}{L}_k^{+}:& =(x-2\eta )\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j\\ & +2\eta \sum _{j=0}^k\sum _{m=0}^j{(-1)}^{j-m}{w}^{-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1},k\ge 1,\end{array}$(50) and (51) $\begin{array}{rl}& [2\eta \sum _{j=0}^k\sum _{m=0}^j{(-1)}^{j-m}{w}^{-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{j+1}\\ & -2\eta \sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}+(x+w)\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j+2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}-(k+1)w]\\ & \times B{l}_k^{\left[2\right]}=0,k\ge 1\end{array}$(51)

where $B{l}_{m,w}$ are the degenerate Boole numbers in (48(48) $\frac{1}{1+{(1+z)}^{\frac{\eta }{w}}}=\sum _{m=0}^{\mathrm{\infty }}B{l}_{m,w}\frac{z^m}{m!}.$(48) ), respectively.

Corollary 4.9

The IPLO, IPRO and IPDE satisfied by the TI degenerate Gould–Hopper Boole polynomials are as follows: (52) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{k}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w\end{array}$(52) (53) $\begin{array}{rl}{L}_k^{+}:& =[(x-2\eta )\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +2\eta \sum _{j=0}^k\sum _{m=0}^j{(-w)}^{j-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}],k\ge 1\end{array}$(53) and (54) $\begin{array}{rl}& [(x-2\eta )\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +2\eta \sum _{j=0}^k\sum _{m=0}^j{(-w)}^{j-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +2(y+w)\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+2}\\ & +2w\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}-(k+1){}_x{\mathrm{\Delta }}_w]\\ & \times B{l}_k^{\left[2\right]}=0,k\ge 1,\end{array}$(54) where $B{l}_{m,w}$ are the degenerate Boole numbers in (48(48) $\frac{1}{1+{(1+z)}^{\frac{\eta }{w}}}=\sum _{m=0}^{\mathrm{\infty }}B{l}_{m,w}\frac{z^m}{m!}.$(48) ), respectively.

4.4. TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials

The TI Gould–Hopper Poisson–Charlier–Boole polynomials ${}_{c}B{l}_k^{\left[2\right]}(x,y;w;\alpha ;\eta ):={}_{c}B{l}_k^{\left[2\right]}$ are defined as follows: (55) $\begin{array}{rl}& {\mathrm{e}}^{-{\alpha }^{w}z}\left(\frac{1}{1+{(1+wz)}^{\frac{\eta }{w}}}\right){(1+wz)}^{\frac{x}{w}}{(1+w{z}^2)}^{\frac{y}{w}}\\ & =\sum _{k=0}^{\mathrm{\infty }}{}_{c}B{l}_k^{\left[2\right]}\frac{z^k}{k!},\end{array}$(55) where $a_1\left(z\right)={\mathrm{e}}^{-{\alpha }^{w}z}\mathrm{a}\mathrm{n}\mathrm{d}{a}_2\left(z\right)=\frac{1}{1+{(1+wz)}^{\frac{\eta }{w}}}.$

Corollary 4.10

The recurrence relation satisfied by the TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials are as follows: (56) $\begin{array}{rl}{}_{c}B{l}_{k+1}^{\left[2\right]}& =-{\alpha }^w{}_{c}B{l}_k^{\left[2\right]}-\eta \sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){(-w)}^{j}j!{}_{c}B{l}_{k-j}^{\left[2\right]}\\ & +\eta \sum _{j=0}^k\sum _{m=0}^j(\genfrac{}{}{0ex}{}{k}{j}){(-w)}^{j-m}j!\frac{B{l}_{m,w}}{m!}{}_{c}B{l}_{k-j}^{\left[2\right]}\\ & +x\sum _{j=0}^k{(-w)}^j\frac{k!}{(k-j)!}{}_{c}B{l}_{k-j}^{\left[2\right]}\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j\frac{k!}{(k-2j-1)!}{}_{c}B{l}_{k-2j-1}^{\left[2\right]},\\ & k\ge 1\end{array}$(56) where $B{l}_{m,w}$ are the degenerate Boole numbers in (48(48) $\frac{1}{1+{(1+z)}^{\frac{\eta }{w}}}=\sum _{m=0}^{\mathrm{\infty }}B{l}_{m,w}\frac{z^m}{m!}.$(48) ), respectively.

Corollary 4.11

The LO, RO and DE satisfied by the TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials are as follows: (57) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{kw}{}_x{\mathrm{\Delta }}_w,\end{array}$(57) (58) $\begin{array}{rl}{L}_k^{+}:& =-{\alpha }^w+(x-\eta )\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j\\ & +\eta \sum _{j=0}^k\sum _{m=0}^j{(-1)}^{j-m}{w}^{-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+1},k\ge 1\end{array}$(58) and (59) $\begin{array}{rl}& [-{\alpha }^w{}_x{\mathrm{\Delta }}_w-\eta \sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +\eta \sum _{j=0}^k\sum _{m=0}^j{(-1)}^{j-m}{w}^{-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{j+1}\\ & +(x+w)\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^{j+1}+w\sum _{j=0}^k{(-1)}^j{}_x{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}\frac{{(-1)}^j}{w^{j+1}}{}_x{\mathrm{\Delta }}_w^{2j+2}-(k+1)w]{}_{c}B{l}_k^{\left[2\right]}=0,\\ & k\ge 1\end{array}$(59) where $B{l}_{m,w}$ are the degenerate Boole numbers in (48(48) $\frac{1}{1+{(1+z)}^{\frac{\eta }{w}}}=\sum _{m=0}^{\mathrm{\infty }}B{l}_{m,w}\frac{z^m}{m!}.$(48) ), respectively.

Corollary 4.12

The IPLO, IPRO and IPDE satisfied by the TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials are as follows: (60) $\begin{array}{rl}{L}_k^{-}:& =\frac{1}{k}{}_x{\mathrm{\Delta }}_w^{-1}{}_y{\mathrm{\Delta }}_w,\end{array}$(60) (61) $\begin{array}{rl}{L}_k^{+}:& =[-{\alpha }^w+(x-\eta )\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +\eta \sum _{j=0}^k\sum _{m=0}^j{(-w)}^{j-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^j\\ & +2y\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}],k\ge 1\end{array}$(61) and (62) $\begin{array}{rl}& [-{\alpha }^w{}_y{\mathrm{\Delta }}_w+(x-\eta )\sum _{j=0}^k{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +\eta \sum _{j=0}^k\sum _{m=0}^j{(-w)}^{j-m}\frac{B{l}_{m,w}}{m!}{}_x{\mathrm{\Delta }}_w^{-j}{}_y{\mathrm{\Delta }}_w^{j+1}\\ & +2(y+w)\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+2}\\ & +2w\sum _{j=0}^{\left[\frac{k-1}{2}\right]}{(-w)}^j{}_x{\mathrm{\Delta }}_w^{-(2j+1)}{}_y{\mathrm{\Delta }}_w^{2j+1}-(k+1){}_x{\mathrm{\Delta }}_w]\\ & \times {}_{c}B{l}_k^{\left[2\right]}=0,k\ge 1\end{array}$(62) where $B{l}_{m,w}$ are Boole numbers in (48(48) $\frac{1}{1+{(1+z)}^{\frac{\eta }{w}}}=\sum _{m=0}^{\mathrm{\infty }}B{l}_{m,w}\frac{z^m}{m!}.$(48) ), respectively.

5. Conclusion

In this paper, we introduce the sequence of TI-DGHAP. We prove an equivalence theorem for TI-DGHAP. We find some of their characteristic properties such as explicit representation, determinantal representation, recurrence relation, LO, RO, DE, IPLO, IPRO and IPDE. We present TI degenerate Gould–Hopper Bernoulli polynomials, TI degenerate Gould–Hopper Poisson–Chalier polynomials, TI degenerate Gould–Hopper Boole polynomials and TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials for special cases of TI-DGHAP.

Corresponding to the same idea, we can define the r-iterated ${\mathrm{\Delta }}_w$-GHAP ${\mathcal{A}}_k^{\left[r\right]}(x,y;w):={\mathcal{A}}_k^{\left[r\right]}$ by the generating function given by (63) $\begin{array}{rl}& [a_r\left(z\right)a_{r-1}\left(z\right)\dots a_3\left(z\right)a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}\\ & \times {(1+w{z}^2)}^{\frac{y}{w}}]=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[r\right]}\frac{z^k}{k!}.\end{array}$(63) The investigation of the polynomials in (63(63) $\begin{array}{rl}& [a_r\left(z\right)a_{r-1}\left(z\right)\dots a_3\left(z\right)a_2\left(z\right)a_1\left(z\right){(1+wz)}^{\frac{x}{w}}\\ & \times {(1+w{z}^2)}^{\frac{y}{w}}]=\sum _{k=0}^{\mathrm{\infty }}{\mathcal{A}}_k^{\left[r\right]}\frac{z^k}{k!}.\end{array}$(63) ) will give rise to many interesting and potentially useful results and left as a future work.

Acknowledgments

We would like to thank the editor and reviewers for their valuable suggestions and comments.

Disclosure statement

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

Additional information

Funding

We would like to thanks the Scientific and Technological Research Council of Türkiye (TÜBİTAK) for the TÜBİTAK BİDEB 2211-A General Domestic Doctorate Scholarship Program that supported the first author.