A New Family of Appell-Type Changhee Polynomials with Geometric Applications

Abstract

Recently, Appell-type polynomials have been investigated and applied in several ways. In this paper, we consider a new extension of Appell-type Changhee polynomials. We introduce two-variable generalized Appell-type $\lambda$-Changhee polynomials (2VGAT$\lambda$CHP). The generating function, series representations, and summation identities related to these polynomials are explored. Further, certain symmetry identities involving two-variable generalized Appell-type $\lambda$-Changhee polynomials are established. Finally, Mathematica was used to examine the zero distributions of two-variable truncated-exponential Appell-type Changhee polynomials.

1. Introduction and Preliminaries

Many special polynomials and numbers have created many spirited applications in several fields such as mathematics, mathematical modeling, statistics, mathematical physics, applied mathematics, and engineering [1,2,3,4,5,6,7,8,9,10]. Recently, Changhee polynomials and numbers have been frequently used and applied in many branches of mathematics, especially in areas such as mathematical physics, mathematical modeling, fractional analysis, and analytical number theory, for example, Adel et al. [1] introduced a numerical investigation for a fractional model of pollution for a system of lakes using the spectral collocation method (SCM) based on Appell-type Changhee polynomials. Adel et al. [2] presented a high-dimensional chaotic Lorenz system, providing a numerical treatment using Appell-type Changhee polynomials. Khater et al. [7] investigated the non-Newtonian nanofluid flow across an exponentially stretching sheet with viscous dissipation, providing a numerical study using an SCM based on Appell–Changhee polynomials.

Let $\mathcal{R}$ be the set of real numbers, $\mathcal{C}$ denotes the set of complex numbers, $\mathcal{Z}$ denotes the set of integer numbers, $\mathcal{N}=\{1,2,3,\dots \}$ and ${\mathcal{N}}_0=\mathcal{N}\cup \left\{0\right\}$, ${\mathcal{Q}}_p$ is the field of p-adic rational numbers and ${\mathcal{C}}_p$ is the completion of the algebraic closure of ${\mathcal{Q}}_p$, where ${\mathcal{Z}}_p$ denotes the ring of p-adic integers and p is a fixed prime number.

The Changhee polynomials (ChP) $\mathfrak{C}{h}_{\gamma }\left(\xi \right)$ [11,12] are defined by

$${\int }_{{\mathcal{Z}}_p}{(1+\omega )}^{\xi +\delta }d{\mu }_{-1}\left(\delta \right)=\frac{2}{2+\omega }{(1+\omega )}^{\xi }=\sum _{\gamma =0}^{\infty }\mathfrak{C}{h}_{\gamma }\left(\xi \right) \frac{{\omega }^{\gamma }}{\gamma !}.$$

(1)

When $\xi =0$, $\mathfrak{C}{h}_{\gamma }\left(0\right):=\mathfrak{C}{h}_{\gamma }$ are called the Changhee numbers.

The Changhee polynomials of order $\nu$ are defined by

$${\left(\frac{2}{2+\omega }\right)}^{\nu }{(1+\omega )}^{\xi }=\sum _{\gamma =0}^{\infty }\mathfrak{C}{h}_{\gamma }^{\left(\nu \right)}\left(\xi \right) \frac{{\omega }^{\gamma }}{\gamma !}.$$

(2)

When $\xi =0$, $\mathfrak{C}{h}_{\gamma }^{\left(\nu \right)}\left(0\right):=\mathfrak{C}{h}_{\gamma }^{\left(\nu \right)}$ are called the Changhee numbers of order $\nu$.

Lee et al. [13] introduced Appell-type Changhee polynomials (ATCHP), which are defined as

$$\frac{2}{2+\omega }{e}^{\xi \omega }=\sum _{\gamma =0}^{\infty }\mathbb{C}{h}_{\gamma }\left(\xi \right) \frac{{\omega }^{\gamma }}{\gamma !}$$

(3)

and have the following explicit relation:

$$\mathbb{C}{h}_{\gamma }\left(\xi \right)=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma } {\xi }^{\gamma -\epsilon },$$

(4)

where $\mathbb{C}{h}_{\gamma }\left(0\right):=\mathbb{C}{h}_{\gamma }$ are the corresponding Appell-type Changhee numbers.

Lim and Qi [14] introduced Appell-type $\lambda$-Changhee polynomials, which are defined by the generating function

$$\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}{e}^{\xi \omega }=\sum _{\gamma =0}^{\infty }\mathbb{C}{h}_{\gamma }\left(\xi \right|\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},\lambda ,\omega \in {\mathcal{C}}_p$$

(5)

and the explicit relation

$$\mathbb{C}{h}_{\gamma }\left(\xi \right|\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\epsilon }\left(\lambda \right){\xi }^{\gamma -\epsilon },$$

(6)

where $\mathbb{C}{h}_{\gamma }\left(0\right|\lambda )=\mathbb{C}{h}_{\gamma }\left(\lambda \right)$ are the corresponding Appell-type $\lambda$-Changhee numbers.

The Appell-type $\lambda$-Changhee polynomials of order $\nu$ (AT$\lambda$CHP) $\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\xi \right|\lambda )$ [14] are defined by

$${\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{\xi \omega }=\sum _{\gamma =0}^{\infty }\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\xi \right|\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},\lambda ,\omega \in {\mathcal{C}}_p$$

(7)

and have the following explicit relation:

$$\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\xi \right|\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}\left(\lambda \right){\xi }^{\gamma -\epsilon },$$

(8)

where $\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(0\right|\lambda )=\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\lambda \right)$ are the corresponding Appell-type $\lambda$-Changhee numbers of order $\nu$. Note that $\mathbb{C}{h}_{\gamma }^{\left(1\right)}\left(0\right|1)=\mathbb{C}{h}_{\gamma }^{\left(1\right)}\left(1\right)=\mathbb{C}{h}_{\gamma }.$

Further, the AT$\lambda$CHP $\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\xi \right|\lambda )$ satisfy the following recurrence relation:

$$\frac{\mathrm{d}}{\mathrm{d}\xi }\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\xi \right|\lambda )=\gamma \mathbb{C}{h}_{\gamma -1}^{\left(\nu \right)}\left(\xi \right|\lambda ).$$

(9)

Due to the importance of the special functions of two variables in applications, a general class of the two-variable polynomials, namely, the two-variable general polynomials (2VGP) ${\mathbb{G}}_{\gamma }(\xi ,\delta )$, is presented in [15]. These polynomials are defined by the generating function

$$e^{\xi \omega }\psi (\delta ,\omega )=\sum _{\gamma =0}^{\infty }{\mathbb{G}}_{\gamma }(\xi ,\delta ) \frac{{\omega }^{\gamma }}{\gamma !}, {\mathbb{G}}_0(\xi ,\delta )=1,$$

(10)

where

$$\psi (\delta ,\omega )=\sum _{\gamma =0}^{\infty }{\psi }_{\gamma }\left(\delta \right) \frac{{\omega }^{\gamma }}{\gamma !}, {\psi }_0\left(\delta \right)\ne 0.$$

(11)

The 2VGP ${\mathbb{G}}_{\gamma }(\xi ,\delta )$ satisfy the following operators:

$${\widehat{M}}_{\mathbb{G}}=\xi +\frac{{\psi }^{\prime }(\delta ,D_{\xi })}{\psi (\delta ,D_{\xi })}\left(D_{\xi }:=\frac{\partial }{\partial \xi };{\psi }^{\prime }(\delta ,\omega ):=\frac{\partial }{\partial \omega }\psi (\delta ,\omega )\right)$$

(12)

and

$${\widehat{P}}_{\mathbb{G}}=D_{\xi },$$

(13)

respectively.

Moreover, the 2VGP ${\mathbb{G}}_{\gamma }(\xi ,\delta )$ meet the following identities:

$$\begin{array}{cc}\hfill & {\widehat{M}}_{\mathbb{G}}\left\{{\mathbb{G}}_{\gamma }(\xi ,\delta )\right\}={\mathbb{G}}_{\gamma +1}(\xi ,\delta ),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\widehat{P}}_{\mathbb{G}}\left\{{\mathbb{G}}_{\gamma }(\xi ,\delta )\right\}=\gamma {\mathbb{G}}_{\gamma -1}(\xi ,\delta ),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\widehat{M}}_{\mathbb{G}}{\widehat{P}}_{\mathbb{G}}\left\{{\mathbb{G}}_{\gamma }(\xi ,\delta )\right\}=\gamma {\mathbb{G}}_{\gamma }(\xi ,\delta ),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & exp\left({\widehat{M}}_{\mathbb{G}}\omega \right)\left\{1\right\}=\sum _{\gamma =0}^{\infty }{\mathbb{G}}_{\gamma }(\xi ,\delta ) \frac{{\omega }^{\gamma }}{\gamma !}\left(\right|\omega |<\infty ).\hfill \end{array}$$

(17)

Some members of the 2VGP ${\mathbb{G}}_{\gamma }(\xi ,\delta )$ are mentioned in Table 1.

Table 1. Some members of 2VGP class ${\mathbb{G}}_{\gamma }(\xi ,\delta )$.

S. No.	$\mathit{\psi }(\mathit{\delta },\mathit{\omega })$	Generating Functions	Polynomials
I.	$e^{\delta {\omega }^r}$	$e^{\xi \omega +\delta {\omega }^r}={\sum }_{\gamma =0}^{\infty }{\mathcal{H}}_{\gamma }^{\left(r\right)}(\xi ,\delta )\frac{{\omega }^{\gamma }}{\gamma !}$	Gould–Hopper polynomials ${\mathbb{H}}_{\gamma }^{\left(r\right)}(\xi ,\delta )$ [16]
II.	$C_0\left(\delta \omega \right)$	$e^{\xi \omega } C_0\left(\delta \omega \right)={\sum }_{\gamma =0}^{\infty }{\mathtt{L}}_{\gamma }(\delta ,\xi )\frac{{\omega }^{\gamma }}{\gamma !}$	Two-variable Laguerre polynomials ${\mathtt{L}}_{\gamma }(\delta ,\xi )$ [17]
III.	$\frac{1}{1-\delta {\omega }^s}$	$\frac{1}{1-\delta {\omega }^s} e^{\xi \omega }={\sum }_{\gamma =0}^{\infty }{e}_{\gamma }^{\left(s\right)}(\xi ,\delta ) \frac{{\omega }^{\gamma }}{\gamma !}$	Two-variable truncated exponential of order s, $e_{\gamma }^{\left(s\right)}(\xi ,\delta )$ [18]
IV.	$A\left(\omega \right) e^{\delta {\omega }^2}$	$A\left(\omega \right) e^{\xi \omega +\delta {\omega }^2}={\sum }_{\gamma =0}^{\infty }{ }_{\mathcal{H}}{A}_{\gamma }(\xi ,\delta ) \frac{{\omega }^{\gamma }}{\gamma !}$	Hermite–Appell polynomials ${ }_{\mathcal{H}}{A}_{\gamma }(\xi ,\delta )$ [19]
V.	$\frac{1}{1-\delta (e^{\omega }-1)}$	$\frac{1}{1-\delta (e^{\omega }-1)} e^{\xi \omega }={\sum }_{\gamma =0}^{\infty }{F}_{\gamma }(\xi ,\delta ) \frac{{\omega }^{\gamma }}{\gamma !}$	Two-variable Fubini polynomials $F_{\gamma }(\xi ,\delta )$ [20,21]

Note that

$${\left(\xi \right)}_{\gamma }=\xi (\xi -1)\dots (\xi -\gamma +1)=\sum _{r=0}^{\gamma }{S}_1(\gamma ,r) {\xi }^r,\gamma \ge 0,$$

(18)

where $S_1(\gamma ,r)$ are called the first-kind Stirling numbers [22] and given by

$$\frac{1}{r!}{(log(1+\omega ))}^r=\sum _{\gamma =r}^{\infty }{S}_1(\gamma ,r)\frac{{\omega }^{\gamma }}{\gamma !}.$$

(19)

The following recurrence relation defines the second-kind Stirling numbers:

$${\xi }^{\gamma }=\sum _{r=0}^{\gamma }{S}_2(\gamma ,r) {\left(\xi \right)}_r,$$

(20)

which also can be given by

$$\frac{1}{r!}{(e^{\omega }-1)}^r=\sum _{\gamma =r}^{\infty }{S}_2(\gamma ,r)\frac{{\omega }^{\gamma }}{\gamma !},r\ge 0.$$

(21)

The generating relation of the sum of integer powers $S_r\left(\gamma \right)={\sum }_{s=0}^{\gamma }{s}^r$ [22] is given by

$$\sum _{r=o}^{\infty }{S}_r\left(\gamma \right)\frac{{\omega }^r}{r!}=\frac{e^{(\gamma +1)\omega }-1}{e^{\omega }-1}.$$

(22)

Due to the exigency of discussing some significant problems in various fields or mathematical interests, recently, a noteworthy number of new generalized and hybrid special polynomials and numbers have been considered [15,19,21,23,24,25,26,27,28,29]. Over the past few years, research on Changhee polynomials and their generalizations has been actively conducted by many researchers. For instance, Kim et al. [12] investigated some results on Changhee numbers and polynomials. Kim et al. [30] introduced higher-order Changhee numbers and polynomials and examined some related properties. Lee et al. [13] established the Appell-type Changhee polynomials and numbers. Lim and Qi [14] introduced Appell-type $\lambda$-Changhee polynomials and explored some related identities. Kim et al. [11] presented a note on nonlinear Changhee differential equations. Pathan and Khan [31] discussed the Appell type $\lambda$-Changhee–Hermite polynomials and their properties. Nahid et al. [32] studied truncated-exponential-based Appell-type Changhee polynomials. Rim et al. [33] defined twisted Changhee polynomials and found some relationships between Euler polynomials, Stirling numbers of the first and the second kind, and these polynomials. Jang et al. [34] investigated twisted Changhee polynomials and numbers and also discussed some properties. Kim et al. [35] introduced Changhee–Genocchi polynomials and numbers and investigated some explicit identities.

Motivated by the above-mentioned works, in this article, over the hybridization of the Appell-type $\lambda$-Changhee polynomials with two-variable general polynomials, we produce a new attractive and useful class of mixed special polynomials called two-variable generalized Appell-type $\lambda$-Changhee polynomials. Additionally, certain series representations and some other properties related to these polynomials are investigated. Next, some symmetry identities involving our produced polynomials are obtained. Further, the zero distributions of certain related members are discussed.

2. Two-Variable Generalized Appell-Type $\mathit{\lambda }$-Changhee Polynomials

In this section, through the help of the monomiality principle, we commingle the AT$\lambda$CHP with the 2VGP to create a class of a more generalized family of hybrid polynomials called two-variable generalized Appell-type $\lambda$-Changhee polynomials by means of a generating function and series definition. Further, we also derive certain properties and summation formulae for these polynomials.

Here, we consider the generating functions (7) and (10), and assume that $\lambda ,\omega \in {\mathcal{C}}_p$ such that ${\left|\lambda \omega \right|}_p<p^{-1/p-1}$. According to the monomiality principle [36,37] and utilizing identities (12) and (17), we define the 2VGAT$\lambda$CHP of order $\nu$ as

$${\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{\xi \omega }\psi (\delta ,\omega )=\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.$$

(23)

Next, we present some special members of the two-variable generalized Appell-type $\lambda$-Changhee family as:

• Choosing $\lambda =1$ in (23) gives the two-variable generalized Appell-type Changhee polynomials of order $\nu$ which are defined by

  $${\left(\frac{2}{2+\omega }\right)}^{\nu }{e}^{\xi \omega }\psi (\delta ,\omega )=\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta ) \frac{{\omega }^{\gamma }}{\gamma !}.$$

  (24)
• Setting $\psi (\delta ,\omega )=\frac{1}{1-\delta {\omega }^s} \mathrm{and} \nu =1$ in (24) gives

  $$\frac{2}{(2+\omega )(1-\delta {\omega }^s)}{e}^{\xi \omega }=\sum _{\gamma =0}^{\infty }{ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }(\xi ,\delta ) \frac{{\omega }^{\gamma }}{\gamma !},$$

  (25)

  where ${ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }(\xi ,\delta )$ are called the two-variable truncated-exponential-Appell-type Changhee polynomials of order s.
• Setting $\delta =s=1$ in (25), we obtain

  $$\frac{2}{(2+\omega )(1-\omega )}{e}^{\xi \omega }=\sum _{\gamma =0}^{\infty }{ }_e\mathbb{C}{h}_{\gamma }\left(\xi \right) \frac{{\omega }^{\gamma }}{\gamma !},$$

  (26)

  where ${ }_e\mathbb{C}{h}_{\gamma }\left(\xi \right)$ are called the truncated-exponential Appell-type Changhee polynomials [32].
• Setting $\psi (\delta ,\omega )=e^{\delta {\omega }^r}$ in (23), we obtain

  $${\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{\xi \omega +\delta {\omega }^r}=\sum _{\gamma =0}^{\infty }{ }_H\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},$$

  (27)

  where ${ }_H\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ are called the Gould–Hopper Appell-type $\lambda$-Changhee polynomials of order $\nu$.
• Setting $\psi (\delta ,\omega )=C_0\left(\delta \omega \right)$ in (23), we obtain

  $${\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{\xi \omega } C_0\left(\delta \omega \right)=\sum _{\gamma =0}^{\infty }{ }_L\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},$$

  (28)

  where ${ }_L\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ are called the two-variable Laguerre Appell-type $\lambda$-Changhee polynomials of order $\nu$.
• Setting $\psi (\delta ,\omega )=\frac{1}{1-\delta {\omega }^s}$ in (23), we obtain

  $${\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }\frac{e^{\xi \omega }}{1-\delta {\omega }^s}=\sum _{\gamma =0}^{\infty }{ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},$$

  (29)

  where ${ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ are called the two-variable truncated-exponential Appell-type $\lambda$-Changhee polynomials (ETAT$\lambda$CHP) of order $\nu$.
• Setting $\psi (\delta ,\omega )=A\left(\omega \right) e^{\delta {\omega }^2}$ in (23), we obtain

  $${\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }A\left(\omega \right) e^{\xi \omega +\delta {\omega }^2}=\sum _{\gamma =0}^{\infty }{ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},$$

  (30)

  where ${ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ are called the Hermite–Appell-based Appell-type $\lambda$-Changhee polynomials of order $\nu$.
• Setting $\psi (\delta ,\omega )=\frac{1}{1-\delta (e^{\omega }-1)}$ in (23), we obtain

  $${\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }\frac{ e^{\xi \omega }}{1-\delta (e^{\omega }-1)}=\sum _{\gamma =0}^{\infty }{ }_F\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},$$

  (31)

  where ${ }_F\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ are called the Fubini Appell-type $\lambda$-Changhee polynomials of order $\nu$.

From (7), (10) and (23), we obtain

$$\begin{array}{cc}\hfill \sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}& ={\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{\xi \omega }\psi (\delta ,\omega )\hfill \\ & =\left(\sum _{\gamma =0}^{\infty }\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\lambda \right) \frac{{\omega }^{\gamma }}{\gamma !}\right)\left(\sum _{\epsilon =0}^{\infty }{\mathbb{G}}_{\epsilon }(\xi ,\delta ) \frac{{\omega }^{\epsilon }}{\epsilon !}\right)\hfill \\ & =\sum _{\gamma =0}^{\infty }\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\lambda \right)\sum _{\epsilon =0}^{\infty }{\mathbb{G}}_{\epsilon }(\xi ,\delta ) \frac{{\omega }^{\gamma +\epsilon }}{\gamma !\epsilon !}\hfill \\ & =\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}\left(\lambda \right) {\mathbb{G}}_{\epsilon }(\xi ,\delta )\frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(32)

From (32), we reach the following theorem.

Theorem 1.

For $\gamma \in {\mathcal{N}}_0$, $\lambda \in {\mathcal{C}}_p$, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}\left(\lambda \right) {\mathbb{G}}_{\epsilon }(\xi ,\delta ).$$

(33)

Similarly, we can obtain

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}\left(\xi \right|\lambda ) {\psi }_{\epsilon }\left(\delta \right);$$

(34)

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(0,\delta |\lambda ) {\xi }^{\gamma -\epsilon }.$$

(35)

From (18), (21) and (23), we obtain

$$\begin{array}{cc}\hfill \sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}& ={\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{\xi \omega }\psi (\delta ,\omega )\hfill \\ & ={\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }\psi (\delta ,\omega ) {(e^{\omega }-1+1)}^{\xi }\hfill \\ & =\left(\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(0,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}\right)\left(\sum _{\epsilon =0}^{\infty }{\left(\xi \right)}_{\epsilon } \frac{1}{\epsilon !}{(e^{\omega }-1)}^{\epsilon }\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\left(\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(0,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}\right)\left(\sum _{\epsilon =0}^{\infty }{\left(\xi \right)}_{\epsilon } \sum _{h=m}^{\infty }{S}_2(h,\epsilon ) \frac{{\omega }^h}{h!}\right)\hfill \\ & =\left(\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(0,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}\right)\left(\sum _{h=0}^{\infty }\sum _{\epsilon =0}^h{\left(\xi \right)}_{\epsilon } S_2(h,\epsilon ) \frac{{\omega }^h}{h!}\right)\hfill \\ & =\sum _{\gamma =0}^{\infty }\sum _{h=0}^{\gamma }\sum _{\epsilon =0}^h\left(\genfrac{}{}{0pt}{}{\gamma }{h}\right){\left(\xi \right)}_{\epsilon } S_2(h,\epsilon ){ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -h}^{\left(\nu \right)}(0,\delta |\lambda )\frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(36)

From (36), we obtain the following theorem.

Theorem 2.

Let $\gamma \in {\mathcal{N}}_0$, $\lambda \in {\mathcal{C}}_p$. Then, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )=\sum _{h=0}^{\gamma }\sum _{\epsilon =0}^h\left(\genfrac{}{}{0pt}{}{\gamma }{h}\right){\left(\xi \right)}_{\epsilon } S_2(h,\epsilon ){ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -h}^{\left(\nu \right)}(0,\delta |\lambda ).$$

(37)

Replacing $\xi$ by z in (23), we have

$$\begin{array}{cc}\hfill \sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(z,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}& ={\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{z\omega }\psi (\delta ,\omega )\hfill \\ & ={\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{(\xi -\theta )\omega } e^{-(\xi -z-\theta )\omega } \psi (\delta ,\omega )\hfill \\ & =e^{(z-\xi +\theta )\omega }\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi -\theta ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}\hfill \\ & =\left(\sum _{\gamma =0}^{\infty }\frac{{(z-\xi +\theta )}^{\gamma } {\omega }^{\gamma }}{\gamma !}\right)\left(\sum _{\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\xi -\theta ,\delta |\lambda ) \frac{{\omega }^{\epsilon }}{\epsilon !}\right)\hfill \\ & =\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){(z-\xi +\theta )}^{\gamma -\epsilon } { }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\xi -\theta ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(38)

From (38), the following theorem is obtained.

Theorem 3.

For $\gamma \in {\mathcal{N}}_0$, $\lambda \in {\mathcal{C}}_p$, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(z,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){(z-\xi +\theta )}^{\gamma -\epsilon } { }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\xi -\theta ,\delta |\lambda ).$$

(39)

In (23), replacing $\xi$ by $\xi +u$ and $\nu$ by $\nu +\zeta$, we have

$$\begin{array}{cc}\hfill \sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{(\nu +\zeta )}(\xi +u,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}& ={\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{(\nu +\zeta )}{e}^{(\xi +u)\omega } \psi (\delta ,\omega )\hfill \\ & =\left(\sum _{\gamma =0}^{\infty }\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}\left(\xi \right|\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}\right)\left(\sum _{\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\zeta \right)}(u,\delta |\lambda ) \frac{{\omega }^{\epsilon }}{\epsilon !}\right)\hfill \\ & =\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}\left(\xi \right|\lambda ) { }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\zeta \right)}(u,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(40)

From (40), the following theorem is obtained.

Theorem 4.

For $\gamma \in {\mathcal{N}}_0$, $\lambda \in {\mathcal{C}}_p$, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{(\nu +\zeta )}(\xi +u,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}\left(\xi \right|\lambda ) { }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\zeta \right)}(u,\delta |\lambda ).$$

(41)

In view of (23), we can write

$$\begin{array}{cc}\hfill \sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\kappa ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}& ={\left(\frac{2}{{(1+\lambda \omega )}^{1/\lambda }+1}\right)}^{\nu }{e}^{(\xi +\kappa )\omega } \psi (\delta ,\omega )\hfill \\ & =\left(\sum _{\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\epsilon }}{\epsilon !}\right)\left(\sum _{\gamma =0}^{\infty } \frac{{\kappa }^{\gamma } {\omega }^{\gamma }}{\gamma !}\right)\hfill \\ & =\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right) { }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\xi ,\delta |\lambda ) {\kappa }^{\gamma -\epsilon } \frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(42)

From (42), we acquire the following theorem.

Theorem 5.

For $\gamma \in {\mathcal{N}}_0$, $\lambda \in {\mathcal{C}}_p$, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\kappa ,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right) { }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\xi ,\delta |\lambda ) {\kappa }^{\gamma -\epsilon }.$$

(43)

Similarly, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\kappa ,\delta |\lambda )=\sum _{l=0}^{\gamma }\sum _{\epsilon =0}^l\left(\genfrac{}{}{0pt}{}{\gamma }{l}\right) {\left(\kappa \right)}_{\epsilon } S_2(l,\epsilon ){ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -l}^{\left(\nu \right)}(\xi ,\delta |\lambda );$$

(44)

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\delta ,\delta |\lambda )=\sum _{l=0}^{\gamma }\sum _{\epsilon =0}^l\left(\genfrac{}{}{0pt}{}{\gamma }{l}\right) {\left(\xi \right)}_{\epsilon } S_2(l+\kappa ,\epsilon +\kappa ){ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -l}^{\left(\nu \right)}(0,\delta |\lambda );$$

(45)

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\kappa ,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\xi ,\delta |\lambda ) {\mathbb{G}}_{\epsilon }(\kappa ,\delta ).$$

(46)

Taking $\kappa =\xi$ in (46), we obtain

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(2\xi ,\delta |\lambda )=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\xi ,\delta |\lambda ) {\mathbb{G}}_{\epsilon }(\xi ,\delta ).$$

(47)

Replacing $\omega$ by $\omega +s$ in (23), we obtain

$$\begin{array}{c}\hfill {\left(\frac{2}{{(1+\lambda (\omega +s))}^{1/\lambda }+1}\right)}^{\nu } \psi (\delta ,\omega +s)=e^{-\xi (\omega +s)}\sum _{\gamma ,\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma +\epsilon }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }{s}^{\epsilon }}{\gamma ! \epsilon !}.\end{array}$$

(48)

Now, replacing $\xi$ by z in Equation (48), comparing the resultant equation with Equation (48) and simplifying, we have

$$\begin{array}{cc}\hfill \sum _{\gamma ,\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma +\epsilon }^{\left(\nu \right)}(z,\delta |\lambda ) \frac{{\omega }^{\gamma }{s}^{\epsilon }}{\gamma ! \epsilon !}& =\sum _{\varrho =0}^{\infty }\frac{{\left((z-\xi )(\omega +s)\right)}^{\varrho }}{\varrho !} \sum _{\gamma ,\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma +\epsilon }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }{s}^{\epsilon }}{\gamma ! \epsilon !}\hfill \\ & =\sum _{\gamma ,\epsilon =0}^{\infty }\sum _{l,k=0}^{\gamma ,\epsilon }\frac{{(z-\xi )}^{l+k} { }_{\mathbb{G}}\mathbb{C}{h}_{\gamma +\epsilon -l-k}^{\left(\nu \right)}(\xi ,\delta |\lambda ) {\omega }^{\gamma }{s}^{\epsilon }}{l! k! (\gamma -l)! (\epsilon -k)!}\hfill \\ & =\sum _{\gamma ,\epsilon =0}^{\infty }\sum _{l,k=0}^{\gamma ,\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{l}\right)\left(\genfrac{}{}{0pt}{}{\epsilon }{k}\right){(z-\xi )}^{l+k} { }_{\mathbb{G}}\mathbb{C}{h}_{\gamma +\epsilon -l-k}^{\left(\nu \right)}(\xi ,\delta |\lambda )\frac{{\omega }^{\gamma }{s}^{\epsilon }}{\gamma ! \epsilon !}.\hfill \end{array}$$

(49)

From (49), the following theorem is obtained.

Theorem 6.

For $\gamma \in {\mathcal{N}}_0$ and $\lambda \in {\mathcal{C}}_p$, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma +\epsilon }^{\left(\nu \right)}(z,\delta |\lambda )=\sum _{l,k=0}^{\gamma ,\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{l}\right)\left(\genfrac{}{}{0pt}{}{\epsilon }{k}\right){(z-\xi )}^{l+k} { }_{\mathbb{G}}\mathbb{C}{h}_{\gamma +\epsilon -l-k}^{\left(\nu \right)}(\xi ,\delta |\lambda ).$$

(50)

Multiplying the right-hand side of (23) by $(e^{\epsilon \omega }+1)$, we have

$$\begin{array}{cc}\hfill (e^{\epsilon \omega }+1)\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}& =e^{\epsilon \omega }\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}+\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}\hfill \\ & =\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\epsilon ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}+\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(51)

From (51), we obtain

$$\begin{array}{cc}\hfill \sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}& =\frac{1}{e^{\epsilon \omega }+1}\left(\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\epsilon ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}+\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}\right)\hfill \\ & =\frac{1}{2}\left(\sum _{k=0}^{\infty }{T}_{k,\epsilon } \frac{{\omega }^k}{k!}\right)\left(\sum _{\gamma =0}^{\infty }\left({ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi +\epsilon ,\delta |\lambda ) +{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )\right) \frac{{\omega }^{\gamma }}{\gamma !}\right)\hfill \\ & =\frac{1}{2}\sum _{\gamma =0}^{\infty }\sum _{k=0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{k}\right) T_{k,\epsilon } \left({ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -k}^{\left(\nu \right)}(\xi +\epsilon ,\delta |\lambda ) +{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -k}^{\left(\nu \right)}(\xi ,\delta |\lambda )\right)\frac{{\omega }^{\gamma }}{\gamma !},\hfill \end{array}$$

(52)

where $T_{k,\epsilon }$ denotes the generalized tangent numbers [38]. From (52), the following theorem is obtained.

Theorem 7.

For $\gamma \in {\mathcal{N}}_0$ and $\lambda \in {\mathcal{C}}_p$, we have

$${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )=\frac{1}{2}\sum _{k=0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{k}\right) T_{k,\epsilon } \left({ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -k}^{\left(\nu \right)}(\xi +\epsilon ,\delta |\lambda ) +{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -k}^{\left(\nu \right)}(\xi ,\delta |\lambda )\right).$$

(53)

3. Symmetry Identities

The importance of the two-variable forms of the special polynomials in applications and the work of Yang [39] and Özarslan [40] on symmetry identities motivate us to consider symmetry identities for more general families. In this section, symmetry identities for the 2VGAT$\lambda$CHP ${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ are derived. Further, by considering different members of the 2VGP ${\mathbb{G}}_{\gamma }(\xi ,\delta )$, the symmetry identities for certain members belonging to this family are also derived.

Let us consider

$$\chi \left(\omega \right)={\left(\frac{4}{({(1+\lambda \alpha \omega )}^{1/\lambda }+1)({(1+\lambda \beta \omega )}^{1/\lambda }+1)}\right)}^{\nu }{e}^{\alpha \beta (\xi +z)\omega }\psi (\alpha \delta ,\beta \omega )\psi (\beta \delta ,\alpha \omega ).$$

(54)

Clearly, $\chi \left(\omega \right)$ is symmetric with respect to the parameters $\alpha$ and $\beta$; therefore, $\chi \left(\omega \right)$ can be written as

$$\begin{array}{cc}\hfill \chi \left(\omega \right)& =\sum _{\gamma =0}^{\infty }{\alpha }^{\gamma }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\sum _{\epsilon =0}^{\infty }{\beta }^{\gamma }{ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ) \frac{{\omega }^{\epsilon }}{\epsilon !}\hfill \\ & =\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(55)

Similarly, we have

$$\begin{array}{c}\hfill \chi \left(\omega \right)=\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\end{array}$$

(56)

From (55) and (56), we reach the following theorem.

Theorem 8.

For $\gamma \in {\mathcal{N}}_0$, $\alpha ,\beta \in \mathcal{N}$ and $\lambda \in {\mathcal{C}}_p$, we have

$$\begin{array}{cc}\hfill \sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon } & { }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda )\hfill \\ & =\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon }{\alpha }^{\epsilon } { }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ).\hfill \end{array}$$

(57)

Next, note that

$$\frac{e^{ab\omega }-1}{e^{b\omega }-1}=\sum _{\zeta =0}^{a-1}{e}^{b\zeta \omega }$$

(58)

and let

$$\varrho \left(\omega \right)={\left(\frac{4}{({(1+\lambda \alpha \omega )}^{1/\lambda }+1)({(1+\lambda \beta \omega )}^{1/\lambda }+1)}\right)}^{\nu }\frac{e^{2\alpha \beta \xi \omega } {(e^{\alpha \beta \omega }-1)}^2}{(e^{\alpha \omega }-1)(e^{\beta \omega }-1)}\psi (\alpha \delta ,\beta \omega )\psi (\beta \delta ,\alpha \omega ).$$

(59)

Obviously, $\varrho \left(\omega \right)$ is symmetric with respect to the parameters $\alpha$ and $\beta$. We can rewrite $\varrho \left(\omega \right)$ as

$$\begin{array}{cc}\hfill \varrho \left(\omega \right)={\left(\frac{2}{({(1+\lambda \alpha \omega )}^{1/\lambda }+1)}\right)}^{\nu }& \frac{e^{\alpha \beta \xi \omega } (e^{\alpha \beta \omega }-1)}{(e^{\alpha \omega }-1)}\psi (\beta \delta ,\alpha \omega )\hfill \\ & \times {\left(\frac{2}{({(1+\lambda \beta \omega )}^{1/\lambda }+1)}\right)}^{\nu }\frac{e^{\alpha \beta \xi \omega } (e^{\alpha \beta \omega }-1)}{(e^{\beta \omega }-1)}\psi (\alpha \delta ,\beta \omega ),\hfill \end{array}$$

(60)

which upon applying identity (58), becomes

$$\begin{array}{cc}\hfill \varrho \left(\omega \right)={\left(\frac{2}{({(1+\lambda \alpha \omega )}^{1/\lambda }+1)}\right)}^{\nu }& e^{\alpha \beta \xi \omega } \sum _{i=0}^{\beta -1}{e}^{\alpha i\omega }\psi (\beta \delta ,\alpha \omega )\hfill \\ & \times {\left(\frac{2}{({(1+\lambda \beta \omega )}^{1/\lambda }+1)}\right)}^{\nu }{e}^{\alpha \beta \xi \omega } \sum _{j=0}^{\alpha -1}{e}^{\beta j\omega }\psi (\alpha \delta ,\beta \omega )\hfill \end{array}$$

$$\begin{array}{cc}\hfill ={\left(\frac{2}{({(1+\lambda \alpha \omega )}^{1/\lambda }+1)}\right)}^{\nu }& \sum _{i=0}^{\beta -1}{e}^{(\beta \xi +i)\alpha \omega } \psi (\beta \delta ,\alpha \omega )\hfill \\ & \times {\left(\frac{2}{({(1+\lambda \beta \omega )}^{1/\lambda }+1)}\right)}^{\nu } \sum _{j=0}^{\alpha -1}{e}^{(\alpha \xi +j)\beta \omega }\psi (\alpha \delta ,\beta \omega )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{i=0}^{\beta -1}\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\beta \xi +i,\beta \delta |\lambda ) \frac{{\left(\alpha \omega \right)}^{\gamma }}{\gamma !} \sum _{j=0}^{\alpha -1}\sum _{\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\alpha \xi +j,\alpha \delta |\lambda ) \frac{{\left(\beta \omega \right)}^{\epsilon }}{\epsilon !}\hfill \\ & =\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\alpha -1}\sum _{i=0}^{\beta -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi +i,\beta \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi +j,\alpha \delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\hfill \end{array}$$

(61)

Similarly, we can obtain

$$\begin{array}{c}\hfill \varrho \left(\omega \right)=\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\beta -1}\sum _{i=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi +i,\alpha \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi +j,\beta \delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !}.\end{array}$$

(62)

From (61) and (62), the following theorem is obtained.

Theorem 9.

For $\gamma \in {\mathcal{N}}_0$, $\alpha ,\beta \in \mathcal{N}$, and $\lambda \in {\mathcal{C}}_p$, we have

$$\begin{array}{c}\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\alpha -1}\sum _{i=0}^{\beta -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi +i,\beta \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi +j,\alpha \delta |\lambda )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\beta -1}\sum _{i=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi +i,\alpha \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi +j,\beta \delta |\lambda ).\hfill \end{array}$$

(63)

Suppose that

$$\begin{array}{cc}\hfill \sigma \left(\omega \right)& ={\left(\frac{4}{({(1+\lambda \alpha \omega )}^{1/\lambda }+1)({(1+\lambda \beta \omega )}^{1/\lambda }+1)}\right)}^{\nu }\frac{e^{\alpha \beta \xi \omega } (e^{\alpha \beta \omega }+1)}{(e^{\alpha \omega }+1)(e^{\beta \omega }+1)}\psi (\alpha \delta ,\beta \omega )\psi (\beta \delta ,\alpha \omega )\hfill \\ & =\frac{1}{2}\sum _{\gamma =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ) \frac{{\left(\alpha \omega \right)}^{\gamma }}{\gamma !}\sum _{i=0}^{\beta -1}{(-1)}^i{e}^{\alpha i\omega }\sum _{j=0}^{\infty }{E}_j\frac{{\left(\beta \omega \right)}^j}{j!}\sum _{\epsilon =0}^{\infty }{ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\alpha \delta |\lambda ) \frac{{\left(\beta \omega \right)}^{\epsilon }}{\epsilon !}\hfill \\ & =\frac{1}{2}\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\beta -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\alpha }^{\gamma -\epsilon }{\beta }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\alpha \delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},\hfill \end{array}$$

(64)

where $E_j$ represents the Euler numbers [41].

Similarly, we can obtain

$$\begin{array}{c}\hfill \sigma \left(\omega \right)=\frac{1}{2}\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\alpha -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\beta }^{\gamma -\epsilon }{\alpha }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\beta \delta |\lambda ) \frac{{\omega }^{\gamma }}{\gamma !},\end{array}$$

(65)

From (64) and (65), the following theorem is obtained.

Theorem 10.

For $\gamma \in {\mathcal{N}}_0$, $\alpha ,\beta \in \mathcal{N}$, and $\lambda \in {\mathcal{C}}_p$, we have

$$\begin{array}{c}\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\beta -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\alpha }^{\gamma -\epsilon }{\beta }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\alpha \delta |\lambda )\hfill \\ \hspace{1em}=\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\alpha -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\beta }^{\gamma -\epsilon }{\alpha }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_{\mathbb{G}}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\beta \delta |\lambda ).\hfill \end{array}$$

(66)

In view of the special cases of the 2VGP ${\mathbb{G}}_{\gamma }(\xi ,\delta )$ given in Table 1, in Section 2 we defined certain members of 2VGAT$\lambda$CHP ${ }_{\mathbb{G}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$, such as two-variable truncated-exponential Appell-type Changhee polynomials of order s, Gould–Hopper Appell-type $\lambda$-Changhee polynomials of order $\nu$, two-variable Laguerre Appell-type $\lambda$-Changhee polynomials of order $\nu$, two-variable truncated-exponential Appell-type $\lambda$-Changhee polynomials of order $\nu$, Hermite–Appell-based Appell-type $\lambda$-Changhee polynomials of order $\nu$, and Fubini Appell-type $\lambda$-Changhee polynomials of order $\nu$. Next, according to the results (9), (10) and (57), the symmetry identities for the corresponding special members can be obtained.

• Symmetry identities for the two-variable truncated-exponential Appell-type Changhee polynomials of order s ${ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }(\xi ,\delta )$ are obtained as

  $$\begin{array}{c}\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon } e^{\left(s\right)}\mathbb{C}{h}_{\gamma -\epsilon }(\beta \xi ,\beta \delta ){ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\epsilon }(\alpha \xi ,\alpha \delta )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma -\epsilon }(\alpha \xi ,\alpha \delta ){ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\epsilon }(\beta \xi ,\beta \delta );\hfill \end{array}$$

  (67)

  $$\begin{array}{c}\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\alpha -1}\sum _{i=0}^{\beta -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma -\epsilon }(\beta \xi +i,\beta \delta ){ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\epsilon }(\alpha \xi +j,\alpha \delta )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\beta -1}\sum _{i=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma -\epsilon }(\alpha \xi +i,\alpha \delta ){ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\epsilon }(\beta \xi +j,\beta \delta );\hfill \end{array}$$

  (68)

  $$\begin{array}{c}\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\beta -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\alpha }^{\gamma -\epsilon }{\beta }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma -\epsilon }(\beta \xi ,\beta \delta ){ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\epsilon }(0,\alpha \delta )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\alpha -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\beta }^{\gamma -\epsilon }{\alpha }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma -\epsilon }(\alpha \xi ,\alpha \delta ){ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\epsilon }(0,\beta \delta ).\hfill \end{array}$$

  (69)
• Symmetry identities for the Gould–Hopper Appell-type $\lambda$-Changhee polynomials of order $\nu$${ }_H\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ are obtained as

  $$\begin{array}{c}\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_H\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_H\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_H\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_H\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda );\hfill \end{array}$$

  (70)

  $$\begin{array}{c}\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\alpha -1}\sum _{i=0}^{\beta -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_H\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi +i,\beta \delta |\lambda ){ }_H\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi +j,\alpha \delta |\lambda )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\beta -1}\sum _{i=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_H\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi +i,\alpha \delta |\lambda ){ }_H\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi +j,\beta \delta |\lambda );\hfill \end{array}$$

  (71)

  $$\begin{array}{c}\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\beta -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\alpha }^{\gamma -\epsilon }{\beta }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_H\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_H\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\alpha \delta |\lambda )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\alpha -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\beta }^{\gamma -\epsilon }{\alpha }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_H\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_H\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\beta \delta |\lambda ).\hfill \end{array}$$

  (72)
• Symmetry identities for the Hermite–Appell-based Appell-type $\lambda$-Changhee polynomials of order $\nu$${ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$.

  $$\begin{array}{c}\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\epsilon =0}^{\gamma }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda );\hfill \end{array}$$

  (73)

  $$\begin{array}{c}\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\alpha -1}\sum _{i=0}^{\beta -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\alpha }^{\gamma -\epsilon } {\beta }^{\epsilon }{ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi +i,\beta \delta |\lambda ){ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\alpha \xi +j,\alpha \delta |\lambda )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\epsilon =0}^{\gamma }\sum _{j=0}^{\beta -1}\sum _{i=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right){\beta }^{\gamma -\epsilon } {\alpha }^{\epsilon }{ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi +i,\alpha \delta |\lambda ){ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(\beta \xi +j,\beta \delta |\lambda );\hfill \end{array}$$

  (74)

  $$\begin{array}{c}\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\beta -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\alpha }^{\gamma -\epsilon }{\beta }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\beta \xi ,\beta \delta |\lambda ){ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\alpha \delta |\lambda )\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\sum _{\gamma =0}^{\infty }\sum _{\epsilon =0}^{\gamma }\sum _{i=0}^{\alpha -1}\sum _{j=0}^{\epsilon }\left(\genfrac{}{}{0pt}{}{\gamma }{\epsilon }\right)\frac{{(-1)}^i{\beta }^{\gamma -\epsilon }{\alpha }^{\epsilon +j}}{j!(\epsilon -j)!}{E}_j{ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\gamma -\epsilon }^{\left(\nu \right)}(\alpha \xi ,\alpha \delta |\lambda ){ }_{{ }_{\mathcal{H}}A}\mathbb{C}{h}_{\epsilon }^{\left(\nu \right)}(0,\beta \delta |\lambda ).\hfill \end{array}$$

  (75)

Similarly, the symmetry identities for the other special members of the two-variable generalized Appell-type $\lambda$-Changhee family can be obtained.

4. Computer Modeling and Zeros

Over the past few years, there has been increasing interest in solving mathematical problems with the aid of computers. The software Mathematica is used to show the behavior of these newly introduced polynomials and to examine the related zeros. This section aims to demonstrate the benefit of using numerical investigation and computer experiments to support theoretical predictions and to discover new interesting patterns of the zeros of the ETAT$\lambda$CHP ${ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ of order $\nu$ for certain values of the indices and parameters.

Upon utilizing (29) and with the help of Mathematica, we can expand ${ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ explicitly. Here, we mention a few members of ${ }_{e^{\left(s\right)}}\mathbb{C}{h}_{\gamma }^{\left(\nu \right)}(\xi ,\delta |\lambda )$ for $\nu =1$, $\lambda =9$, and $s=3$:

$$\begin{array}{cc}\hfill { }_{e^{\left(3\right)}}\mathbb{C}{h}_0(\xi ,\delta |9)& =1,\hfill \\ \hfill { }_{e^{\left(3\right)}}\mathbb{C}{h}_1(\xi ,\delta |9)& =-\frac{1}{2}+\xi ,\hfill \\ \hfill { }_{e^{\left(3\right)}}\mathbb{C}{h}_2(\xi ,\delta |9)& =\frac{9}{2}-\xi +{\xi }^2,\hfill \\ \hfill { }_{e^{\left(3\right)}}\mathbb{C}{h}_3(\xi ,\delta |9)& =-\frac{323}{4}+\frac{27\xi }{2}-\frac{3{\xi }^2}{2}+{\xi }^3+6\delta ,\hfill \\ \hfill { }_{e^{\left(3\right)}}\mathbb{C}{h}_4(\xi ,\delta |9)& =\frac{4347}{2}-323\xi +27{\xi }^2-2{\xi }^3+{\xi }^4-12\delta +24\xi \delta ,\hfill \\ \hfill { }_{e^{\left(3\right)}}\mathbb{C}{h}_5(\xi ,\delta |9)& =\frac{1}{4}(-312095+43470\xi -3230{\xi }^2+180{\xi }^3-10{\xi }^4+4{\xi }^5+1080\delta \hfill \\ & -240\xi \delta +240{\xi }^2\delta ),\hfill \end{array}$$

$$\begin{array}{cc}\hfill { }_{e^{\left(3\right)}}\mathbb{C}{h}_6(\xi ,\delta |9)& =\frac{1}{4}(14008005-1872570\xi +130410{\xi }^2-6460{\xi }^3+270{\xi }^4\hfill \\ & -12{\xi }^5+4{\xi }^6-38760\delta +6480\xi \delta -720{\xi }^2\delta +480{\xi }^3\delta +2880{\delta }^2).\hfill \end{array}$$

(76)

Approximate solutions satisfying the ETAT$\lambda$CHP ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9)=0$, for certain values, i.e., for $\gamma =5,10,15,20,25,30,35$, and 40 are investigated and presented in Figure 1.

Figure 1. Zero distribution of ETAT$\lambda$CHP ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9)$, for $\gamma =5,10,15,20,25,30,35$, and 40.

We note that the ETAT$\lambda$CHP ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9)$ of degree $\gamma$ has $\gamma$ zeros with the following properties:

• If $\gamma$ is odd, the ETAT$\lambda$CHP ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9)$ has one real zero and $(\gamma -1)$ complex zeros.
• If $\gamma$ is even, the ETAT$\lambda$CHP ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9)$ has $\gamma$ complex zeros.
• The zeros of the ETAT$\lambda$CHP ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9)$ are symmetric with respect to the real axis.

The stacks of zeros of ETAT$\lambda$CHP ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9)$ for $1\le \gamma \le 40$ of 3D form are presented in Figure 2.

Figure 2. Stacks of zeros of ${ }_{e^{\left(3\right)}}\mathbb{C}{h}_{\gamma }(\xi ,15|9),1\le \gamma \le 40$.

5. Conclusions

Recently, the hybrid versions of special polynomials and numbers have been studied by numerous researchers [15,19,21,23,25,26,27,28,29]. In this paper, we established a new generalized class of hybrid special polynomials called two-variable generalized Appell-type $\lambda$-Changhee polynomials. We defined the generating function and derived some helpful series representations for these polynomials. In addition, we investigated some related summations and symmetry identities. Further, we presented some considerable special cases of the two-variable generalized Appell-type $\lambda$-Changhee family, such as the two-variable truncated-exponential-Appell-type Changhee polynomials, Gould–Hopper Appell-type $\lambda$-Changhee polynomials, two-variable Laguerre Appell-type $\lambda$-Changhee polynomials, two-variable truncated-exponential Appell-type $\lambda$-Changhee polynomials, Hermite–Appell-based Appell-type $\lambda$-Changhee polynomials of order $\nu$, and Fubini Appell-type $\lambda$-Changhee polynomials. Moreover, we examined the zeros of two-variable truncated-exponential Appell-type $\lambda$-Changhee polynomials and discussed the behavior of these zeros. We obtained that the zeros of these polynomials are distributed uniformly, and symmetric with respect to the real axis; see Figure 1. The stacked plot of the distribution of the zeros can be also viewed in Figure 2. Our established class is a generalization of many other published forms related to Changhee polynomials. The degenerate form of the generalized Appell-type $\lambda$-Changhee polynomials and related applications can be considered in further studies.