A Note on Higher Order Degenerate Changhee–Genocchi Numbers and Polynomials of the Second Kind

Abstract

In this paper, we consider the higher order degenerate Changhee–Genocchi polynomials of the second kind by using generating functions and the Riordan matrix methods. At the same time, we give some properties of the higher order degenerate Changhee–Genocchi polynomials of the second kind. In addition, we establish some new equalities involving the higher order degenerate Changhee–Genocchi polynomials of the second kind, the generalized Bell polynomials, higher order Changhee polynomials, the higher order degenerate Daehee polynomials of the second kind, Lah numbers and Stirling numbers, etc.

1. Introduction

At first, in 1979, Carlitz studied and discovered about degenerate combinatorial numbers and polynomials, and named degenerate Bernoulli and Euler numbers and polynomials ([1,2]). In recent years, the classical combinatorial numbers and polynomials have been extended to different degrees by using the generating function method, the Riordan array method, and the asymptotic counting method, the probabilistic method, umbral calculus, and so on. Through these methods, the basic properties of combinatorial sequences can be studied, as well as some symmetric forms, the simplification of multiple sums to single sums, etc. In 2009, Taekyun Kim gave some properties of Genocchi numbers and polynomials ([3]). More degenerate versions of classical combinatorial numbers and polynomials are discovered and derived. In 2017, Kim et al. studied the degenerate Changhee–Genocchi polynomials and numbers of the second kind and gave some new identities ([4]). In this paper, we derive some identities involving the higher order degenerate Changhee–Genocchi polynomials and numbers of the second kind and special combination sequences.

The higher order degenerate Changhee–Genocchi polynomials of the second kind are defined by the following generating function in ([4]).

$$\begin{array}{c}\hfill {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{\left(1+\lambda log(1+t)\right)}^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.(\lambda \in \mathbb{R})\end{array}$$

(1)

When $r=1$, $\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}{\left(1+\lambda log(1+t)\right)}^{\frac{x}{\lambda }}={\sum }_{n=0}^{\infty }{J}_{n,\lambda }\left(x\right)\frac{t^n}{n!}$ are called the degenerate Changhee–Genocchi polynomials of the second kind.

The generating functions of the relevant special combinatorial sequences involved in this paper are as follows: (see [1,2,3,5,6,7,8,9,10,11,12,13,14])

The higher order degenerate Euler polynomials are defined by the following generating function to be

$$\begin{array}{c}\hfill {\left(\frac{2}{{\left(1+\lambda t\right)}^{\frac{1}{\lambda }}+1}\right)}^r{\left(1+\lambda t\right)}^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }{\epsilon }_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.(\lambda \in \mathbb{R})\end{array}$$

(2)

The negative order degenerate Euler polynomials are defined by the following generating function to be

$$\begin{array}{c}\hfill {\left(\frac{{\left(1+\lambda t\right)}^{\frac{1}{\lambda }}+1}{2}\right)}^r{\left(1+\lambda t\right)}^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }{\epsilon }_{n,\lambda }^{(-r)}\left(x\right)\frac{t^n}{n!}.(\lambda \in \mathbb{R})\end{array}$$

(3)

The higher order degenerate Genocchi polynomials are defined by the following generating function to be

$$\begin{array}{c}\hfill {\left(\frac{2t}{{\left(1+\lambda t\right)}^{\frac{1}{\lambda }}+1}\right)}^r{\left(1+\lambda t\right)}^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }{G}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.(\lambda \in \mathbb{R})\end{array}$$

(4)

The higher order degenerate Changhee polynomials of the second kind are defined by the following generating function to be

$$\begin{array}{c}\hfill {\left(\frac{2}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{\left(1+\lambda log(1+t)\right)}^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }C{h}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.(\lambda \in \mathbb{R})\end{array}$$

(5)

The higher order degenerate Daehee polynomials of the second kind are defined by the following generating function to be

$$\begin{array}{c}\hfill {\left(\frac{log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}-1}\right)}^r{\left(1+\lambda log(1+t)\right)}^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }{D}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.(\lambda \in \mathbb{R})\end{array}$$

(6)

The higher order Changhee polynomials are defined by

$$\begin{array}{c}\hfill {\left(\frac{2}{2+t}\right)}^k{(1+t)}^x=\sum _{n=0}^{\infty }C{h}_n^{\left(k\right)}\left(x\right)\frac{t^n}{n!}.\end{array}$$

(7)

The higher order Changhee–Genocchi polynomials are defined by

$$\begin{array}{c}\hfill {\left(\frac{2log(1+t)}{2+t}\right)}^k{(1+t)}^x=\sum _{n=0}^{\infty }C{G}_n^{\left(k\right)}\left(x\right)\frac{t^n}{n!}.\end{array}$$

(8)

When $x=0$, $C{G}_n^{\left(k\right)}=C{G}_n^{\left(k\right)}\left(0\right)$ are called the higher order Changhee–Genocchi numbers.

The Lah numbers are given by the generating function

$$\begin{array}{c}\hfill \frac{{\left(\frac{-t}{1+t}\right)}^k}{k!}=\sum _{n\ge k}L(n,k)\frac{t^n}{n!}.\end{array}$$

(9)

The generalized Bell polynomials of the first kind are given by the generating function

$$\begin{array}{c}\hfill \frac{{(e^{e^t-1}-1)}^k}{k!}=\sum _{n=1}^{\infty }B(n,k)\frac{t^n}{n!}.\end{array}$$

(10)

The generalized Bell polynomials of the second kind are given by the generating function

$$\begin{array}{c}\hfill \frac{(log{(1+\left(log(1+t)\right))}^k}{k!}=\sum _{n=1}^{\infty }\beta (n,k)\frac{t^n}{n!}.\end{array}$$

(11)

The Stirling numbers of the first kind and the second kind are defined by

$$\begin{array}{c}\hfill \frac{lo{g}^k(1+t)}{k!}\sum _{n\ge k}{S}_1(n,k)\frac{t^n}{n!}.\end{array}$$

(12)

$$\begin{array}{c}\hfill \frac{{(e^t-1)}^k}{k!}=\sum _{n\ge k}{S}_2(n,k)\frac{t^n}{n!}.\end{array}$$

(13)

The generalized Stirling numbers of the first kind and the second kind are defined by

$$\begin{array}{c}\hfill \frac{lo{g}^k(1+t)}{{(1+t)}^{r}k!}=\sum _{n\ge k}{S}_1(n,k;r)\frac{t^n}{n!}.\end{array}$$

(14)

$$\begin{array}{c}\hfill \frac{{(e^t-1)}^k}{k!}{e}^{rt}=\sum _{n\ge k}{S}_2(n,k;r)\frac{t^n}{n!}.\end{array}$$

(15)

Lemma 1

(Inversion Formula [13]). Let $f,g$ be functions defined on the set of positive integers, then

$$\begin{array}{c}\hfill g_n=\sum _{k=0}^n{S}_1(n,k)f_k\iff f_n=\sum _{k=0}^n{S}_2(n,k)g_k.\end{array}$$

(16)

$$\begin{array}{c}\hfill g_n=\sum _{k=0}^n{S}_1(n,k;r)f_k\iff f_n=\sum _{k=0}^n{S}_2(n,k;r)g_k.\end{array}$$

(17)

A Riordan array is a pair $\left(g\right(t),f(t\left)\right)$ of formal power series with $f_0=f\left(0\right)=0$. It defines an infinite lower triangular array ${\left(d_{n,k}\right)}_{n,k\in \mathbb{N}}$($\mathbb{N}$ is non-negative integer) according to the rule:

$$\begin{array}{c}\hfill d_{n,k}=\left[t^n\right]g\left(t\right){\left(f\left(t\right)\right)}^k.\end{array}$$

(18)

Hence, we write $R\left(d_{n,k}\right)=(g\left(t\right),f\left(t\right))$.

Lemma 2

([14]). If $D=(g\left(t\right),f\left(t\right))={\left(d_{n,k}\right)}_{n,k\in \mathbb{N}}$ is an Riordan array and $h\left(t\right)$ is the generating function of the sequence ${\left(h_k\right)}_{k\in \mathbb{N}}$, i.e, $f\left(t\right)={\sum }_{k=0}^{\infty }{f}_k{t}^k$ or $h\left(t\right)=G\left(h_k\right)$. Then, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n{d}_{n,k}{h}_k=\left[t^n\right]g\left(t\right)h\left(f\left(t\right)\right).\end{array}$$

(19)

2. Some Properties about the Higher Order Degenerate Changhee–Genocchi Numbers and Polynomials of the Second Kind

In this section, we establish some identities and give some properties of the higher order degenerate Changhee–Genocchi numbers and polynomials of the second kind by using generating functions.

Theorem 1.

For non-negative integer n, we obtain

$$\begin{array}{c}\hfill \sum _{n_1+n_2+\cdots n_m=n}\left(\genfrac{}{}{0pt}{}{n}{n_1,n_2,\cdots ,n_m}\right)J_{n_1,\lambda }^{\left(r_1\right)}\left(x_1\right)J_{n_2,\lambda }^{\left(r_2\right)}\left(x_2\right)\cdots J_{n_m,\lambda }^{\left(r_m\right)}\left(x_m\right)=J_{n,\lambda }^{(r_1+\cdots +r_m)}(x_1+\cdots +x_m).\end{array}$$

(20)

Proof. 

By $\left(1\right)$, we get

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{J}_{n,\lambda }^{(r_1+\cdots +r_m)}(x_1+\cdots +x_m)\frac{t^n}{n!}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^{r_1+r_2+\cdots +r_m}{(1+\lambda log(1+t))}^{\frac{x_1+x_2+\cdots +x_m}{\lambda }}\hfill \\ \hfill =& \sum _{n_1=0}^{\infty }{J}_{n_1,\lambda }^{\left(r_1\right)}\left(x_1\right)\frac{t^{n_1}}{n_1!}\cdots \sum _{n_m=0}^{\infty }{J}_{n_m,\lambda }^{\left(r_m\right)}\left(x_m\right)\frac{t^{n_m}}{n_m!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }\sum _{n_1+n_2+\cdots +n_m=n}\left(\genfrac{}{}{0pt}{}{n}{n_1,n_2,\cdots ,n_m}\right)J_{n_1,\lambda }^{\left(r_1\right)}\left(x_1\right)\cdots J_{n_m,\lambda }^{\left(r_m\right)}\left(x_m\right)\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ in both sides, we get the identities. □

Corollary 1.

For $x_i=0(i=1,2,\cdots )$ in $\left(20\right)$, we obtain

$$\begin{array}{c}\hfill \sum _{n_1+n_2+\cdots +n_m=n}\left(\genfrac{}{}{0pt}{}{n}{n_1,n_{2,},\cdots ,n_m}\right)J_{n_1,\lambda }^{\left(r_1\right)}{J}_{n_2,\lambda }^{\left(r_2\right)}\cdots J_{n_m,\lambda }^{\left(r_m\right)}=J_{n,\lambda }^{(r_1+r_2\cdots +r_m)}.\end{array}$$

(21)

Corollary 2.

For $m=2$ in $\left(20\right)$, we obtain

$$\begin{array}{c}\hfill \sum _{m=0}^n\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{m,\lambda }^{\left(r\right)}\left(x\right)J_{n-m,\lambda }^{\left(s\right)}\left(y\right)=J_{n,\lambda }^{(r+s)}(x+y).\end{array}$$

(22)

Corollary 3.

For $s=0$ in $\left(22\right)$, we obtain

$$\begin{array}{c}\hfill \sum _{m=0}^n\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{m,\lambda }^{\left(r\right)}\left(x\right){\left(y\right)}_{n-m,\lambda }=J_{n,\lambda }^{\left(r\right)}(x+y).\end{array}$$

(23)

For $y=0$ in $\left(23\right)$, we obtain

$$\begin{array}{c}\hfill \sum _{m=0}^n\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{m,\lambda }^{\left(r\right)}{\left(x\right)}_{n-m,\lambda }=J_{n,\lambda }^{\left(r\right)}\left(x\right).\end{array}$$

(24)

For $r=0$ in $\left(23\right)$, we obtain

$$\begin{array}{c}\hfill \sum _{m=0}^n\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{m,\lambda }{(x+y)}_{n-m,\lambda }=J_{n,\lambda }(x+y).\end{array}$$

(25)

Theorem 2.

For integer $n\ge k\ge 1$, we have

$$\begin{array}{c}\hfill \frac{d^k\left(\frac{1}{k!}{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\right)}{d{x}^k}=\sum _{m=k}^n\sum _{i=k}^m\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{n-m,\lambda }^{\left(r\right)}\left(x\right){\lambda }^i{S}_1(i,k)S_1(m,i).\end{array}$$

(26)

Proof. 

From generating function $\left(1\right)$, we have

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\frac{d^k\left(\frac{1}{k!}{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\right)}{d{x}^k}\frac{t^n}{n!}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\frac{lo{g}^k(1+\lambda log(1+t))}{k!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\sum _{m=k}^{\infty }{S}_1(m,k){\lambda }^m\frac{{log(1+t)}^m}{m!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\sum _{m=k}^{\infty }{S}_1(m,k){\lambda }^m\sum _{n=m}^{\infty }{S}_1(n,m)\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\sum _{n=k}^{\infty }\sum _{m=k}^n{\lambda }^m{S}_1(m,k)S_1(n,m)\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }\left(\sum _{m=k}^n\sum _{i=k}^m\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{n-m,\lambda }^{\left(r\right)}\left(x\right){\lambda }^i{S}_1(i,k)S_1(m,i)\right)\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ in both sides, we can easily get the identities. □

Corollary 4.

For $k=1$, we have

$$\begin{array}{c}\hfill \frac{d\left(J_{n,\lambda }^{\left(r\right)}\left(x\right)\right)}{dx}=\sum _{m=1}^n\sum _{i=1}^m\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{n-m,\lambda }^{\left(r\right)}\left(x\right){\lambda }^i{(-1)}^{i-1}(i-1)!S_1(m,i).\end{array}$$

(27)

Theorem 3.

For integer $n\ge 1$, we have

$$\begin{array}{c}\hfill \sum _{m=0}^{n-1}\frac{{(-1)}^{n-m-1}}{n-m}{\left(n\right)}_{n-m}{J}_{m,\lambda }^{(r-1)}\left(x\right)=\frac{J_{n,\lambda }^{\left(r\right)}(x+1)+J_{n,\lambda }^{\left(r\right)}\left(x\right)}{2}.\end{array}$$

(28)

Proof. 

From generating function $\left(1\right)$, we have

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }(J_{n,\lambda }^{\left(r\right)}(x+1)+J_{n,\lambda }^{\left(r\right)}\left(x\right))\frac{t^n}{n!}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x+1}{\lambda }}+{\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}[{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1]\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^{r-1}{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\left[2log(1+t)\right]\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{(r-1)}\left(x\right)\frac{t^n}{n!}\sum _{n=1}^{\infty }2{(-1)}^{n-1}(n-1)!\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=1}^{\infty }\sum _{m=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{m,\lambda }^{(r-1)}\left(x\right)2{(-1)}^{n-m-1}(n-m-1)!\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=1}^{\infty }\sum _{m=0}^{n-1}\frac{{(-1)}^{n-m-1}2}{n-m}{\left(n\right)}_{n-m}{J}_{m,\lambda }^{(r-1)}\left(x\right)\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ in both sides, we can easily get the identities. □

Corollary 5.

For $x=0$ in $\left(28\right)$, we have

$$\begin{array}{c}\hfill \sum _{m=0}^{n-1}\frac{{(-1)}^{n-m-1}}{n-m}{\left(n\right)}_{n-m}{J}_{m,\lambda }^{(r-1)}\left(0\right)=\frac{J_{n,\lambda }^{\left(r\right)}\left(1\right)+J_{n,\lambda }^{\left(r\right)}\left(0\right)}{2}.\end{array}$$

(29)

Corollary 6.

For $r=1$ in $\left(29\right)$, we have

$$\begin{array}{c}\hfill \sum _{m=0}^{n-1}\frac{{(-1)}^{n-m-1}}{n-m}{\left(n\right)}_{n-m}=\frac{J_{n,\lambda }\left(1\right)+J_{n,\lambda }\left(0\right)}{2}.\end{array}$$

(30)

Theorem 4.

For non-negative integer n, we obtain

$$\begin{array}{c}\hfill \sum _{m=1}^r\sum _{j=0}^n\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{n}{j}\right)\left(\genfrac{}{}{0pt}{}{r}{m}\right){\left(m\right)}_{l,\lambda }{S}_1(j,l)J_{n-j,\lambda }^{\left(r\right)}+J_{n,\lambda }^{\left(r\right)}=2^{r}r!S_1(n,r).\end{array}$$

(31)

Proof. 

From generating function $\left(1\right),\left(12\right)$, we get

On the one hand,

$$\begin{array}{cc}\hfill 2^{r}lo{g}^r(1+t)=& {\left({\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1\right)}^r\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}+\sum _{m=1}^r\left(\genfrac{}{}{0pt}{}{r}{m}\right){\left(1+\lambda log(1+t)\right)}^{\frac{m}{\lambda }}\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}+\sum _{m=1}^r\left(\genfrac{}{}{0pt}{}{r}{m}\right)\sum _{l=0}^{\infty }\left(\genfrac{}{}{0pt}{}{\frac{m}{\lambda }}{l}\right){\lambda }^{l}lo{g}^l(1+t)\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}+\sum _{m=1}^r\left(\genfrac{}{}{0pt}{}{r}{m}\right)\sum _{l=0}^{\infty }{\left(m\right)}_{l,\lambda }\sum _{j=l}^{\infty }{S}_1(j,l)\frac{t^j}{j!}\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}+\sum _{m=1}^r\left(\genfrac{}{}{0pt}{}{r}{m}\right)\sum _{j=0}^{\infty }\sum _{l=0}^j{\left(m\right)}_{l,\lambda }{S}_1(j,l)\frac{t^j}{j!}\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}+\sum _{m=1}^r\left(\genfrac{}{}{0pt}{}{r}{m}\right)\sum _{n=0}^{\infty }\sum _{j=0}^n\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{n}{j}\right){\left(m\right)}_{l,\lambda }{S}_1(j,l)J_{n-j,\lambda }^{\left(r\right)}\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}+\sum _{m=1}^r\sum _{n=0}^{\infty }\sum _{j=0}^n\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{n}{j}\right)\left(\genfrac{}{}{0pt}{}{r}{m}\right){\left(m\right)}_{l,\lambda }{S}_1(j,l)J_{n-j,\lambda }^{\left(r\right)}\frac{t^n}{n!}.\hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill & 2^{r}lo{g}^r(1+t)=2^{r}r!\frac{lo{g}^r(1+t)}{r!}=\sum _{n=r}^{\infty }{2}^{r}r!S_1(n,r)\frac{t^n}{n!}.\hfill \end{array}$$

Therefore, we obtain the result.

For $r=1$ in $\left(31\right)$, we can get the conclusion from [3]:

$$\begin{array}{c}\hfill \sum _{j=0}^n\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{n}{j}\right){\left(1\right)}_{l,\lambda }{S}_1(j,l)J_{n-j,\lambda }+J_{n,\lambda }=2{(-1)}^{n-1}(n-1)!.\end{array}$$

(32)

□

Theorem 5.

For non-negative integer $n\ge 1$, we have

$$\begin{array}{c}\hfill \sum _{m=1}^{n-r}\left(\genfrac{}{}{0pt}{}{n}{m}\right)C{h}_{n-m,\lambda }^{\left(r\right)}\left(x\right)S_1(m,r)=\frac{J_{n,\lambda }^{\left(r\right)}\left(x\right)}{r!}.\end{array}$$

(33)

Proof. 

From generating function (1), we have

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\hfill \\ \hfill =& {\left(\frac{2}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}lo{g}^r(1+t)\hfill \\ \hfill =& \sum _{n=0}^{\infty }C{h}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\frac{lo{g}^r(1+t)}{r!}r!\hfill \\ \hfill =& \sum _{n=0}^{\infty }C{h}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\sum _{n=r}^{\infty }{S}_1(n,r)\frac{t^n}{n!}r!\hfill \\ \hfill =& \sum _{n=r}^{\infty }\sum _{m=1}^{n-r}r!\left(\genfrac{}{}{0pt}{}{n}{m}\right)C{h}_{n-m,\lambda }^{\left(r\right)}\left(x\right)S_1(m,r)\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ in both sides, we obtain the result. □

Corollary 7.

For $n\ge 1,r=1$ in $\left(33\right)$, we have

$$\begin{array}{c}\hfill \sum _{m=1}^{n-r}\left(\genfrac{}{}{0pt}{}{n}{m}\right)C{h}_{n-m,\lambda }\left(x\right){(-1)}^{m-1}(m-1)!=J_{n,\lambda }\left(x\right).\end{array}$$

(34)

Theorem 6.

For non-negative integer $n\ge r$, we have

$$\begin{array}{c}\hfill \sum _{m=0}^{n-r}\sum _{l=0}^m\left(\genfrac{}{}{0pt}{}{n-r}{m}\right){\epsilon }_{l,\lambda }^{\left(r\right)}\left(x\right)S_1(m,l)D_{n-m}^{\left(r\right)}\left(x\right)=\frac{J_{n,\lambda }^{\left(r\right)}\left(x\right)}{{\left(n\right)}_r}.\end{array}$$

(35)

Proof. 

From generating function $\left(2\right),\left(12\right)$, we get

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\hfill \\ \hfill =& {\left(\frac{2}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\frac{lo{g}^r(1+t)}{t^r}{t}^r\hfill \\ \hfill =& \sum _{l=0}^{\infty }{\epsilon }_{l,\lambda }^{\left(r\right)}\left(x\right)\frac{lo{g}^l(1+t)}{l!}\sum _{n=0}^{\infty }{D}_n^{\left(r\right)}\left(x\right)\frac{t^n}{n!}{t}^r\hfill \\ \hfill =& \sum _{n=0}^{\infty }\sum _{l=0}^n{\epsilon }_{l,\lambda }^{\left(r\right)}\left(x\right)S_1(n,l)\frac{t^n}{n!}\sum _{n=0}^{\infty }{D}_n^{\left(r\right)}\left(x\right)\frac{t^n}{n!}{t}^r\hfill \\ \hfill =& \sum _{n=0}^{\infty }\sum _{m=0}^n\sum _{l=0}^m\left(\genfrac{}{}{0pt}{}{n}{m}\right){\epsilon }_{l,\lambda }^{\left(r\right)}\left(x\right)S_1(m,l)D_{n-m}^{\left(r\right)}\left(x\right)\frac{t^{n+r}}{n!}\hfill \\ \hfill =& \sum _{n=r}^{\infty }\sum _{m=0}^{n-r}\sum _{l=0}^m\left(\genfrac{}{}{0pt}{}{n-r}{m}\right){\left(n\right)}_r{\epsilon }_{l,\lambda }^{\left(r\right)}\left(x\right)S_1(m,l)D_{n-m}^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ in both sides, we can easily get the identities. □

Corollary 8.

For $r=1$ in $\left(35\right)$, we have

$$\begin{array}{c}\hfill \sum _{m=0}^{n-1}\sum _{l=0}^m\left(\genfrac{}{}{0pt}{}{n-1}{m}\right){\epsilon }_{l,\lambda }\left(x\right)S_1(m,l)D_{n-m}\left(x\right)=\frac{J_{n,\lambda }\left(x\right)}{n}.\end{array}$$

(36)

Theorem 7.

For non-negative integer $n\ge r$, we have

$$\begin{array}{c}\hfill \sum _{m=0}^{n-r}\sum _{j=0}^m\left(\genfrac{}{}{0pt}{}{n}{m}\right)r!{\left(2x\right)}_{j,\lambda }{S}_1(m,j)S_1(n-m,r)=\sum _{m=0}^n\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{n-m,\lambda }^{\left(r\right)}\left(x\right){\epsilon }_{m,\lambda }^{(-r)}\left(x\right).\end{array}$$

(37)

Proof. 

From generating function (1), (3), we get

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\sum _{m=0}^n\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{n-m,\lambda }^{\left(r\right)}\left(x\right){\epsilon }_{m,\lambda }^{(-r)}\left(x\right)\frac{t^n}{n!}=\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{\epsilon }_{n,\lambda }^{(-r)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}{\left(\frac{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}{2}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\hfill \\ \hfill =& r!\frac{lo{g}^r(1+t)}{r!}{(1+\lambda log(1+t))}^{\frac{2x}{\lambda }}\hfill \\ \hfill =& r!\sum _{n=r}^{\infty }{S}_1(n,r)\frac{t^n}{n!}\sum _{j=0}^{\infty }{\left(2x\right)}_{j,\lambda }\frac{lo{g}^j(1+t)}{j!}\hfill \\ \hfill =& r!\sum _{n=r}^{\infty }{S}_1(n,r)\frac{t^n}{n!}\sum _{j=0}^{\infty }{\left(2x\right)}_{j,\lambda }\sum _{n=j}^{\infty }{S}_1(n,j)\frac{t^n}{n!}\hfill \\ \hfill =& r!\sum _{n=r}^{\infty }{S}_1(n,r)\frac{t^n}{n!}\sum _{n=0}^{\infty }\sum _{j=0}^n{\left(2x\right)}_{j,\lambda }{S}_1(n,j)\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=r}^{\infty }\sum _{m=0}^{n-r}\sum _{j=0}^m\left(\genfrac{}{}{0pt}{}{n}{m}\right)r!{\left(2x\right)}_{j,\lambda }{S}_1(m,j)S_1(n-m,r)\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ in both sides, we can easily get the identities. □

Corollary 9.

For $r=1$ in $\left(37\right)$, we have

$$\begin{array}{c}\hfill \sum _{m=0}^{n-1}\sum _{j=0}^m\left(\genfrac{}{}{0pt}{}{n}{m}\right){\left(2x\right)}_{j,\lambda }{S}_1(m,j){(-1)}^{n-m-1}(n-m-1)!=\sum _{m=0}^n\left(\genfrac{}{}{0pt}{}{n}{m}\right)J_{n-m,\lambda }\left(x\right){\epsilon }_{m,\lambda }^{(-1)}\left(x\right).\end{array}$$

(38)

Theorem 8.

For non-negative integer n, we have

$$\begin{array}{c}\hfill \sum _{m=0}^n\sum _{j=0}^m\sum _{l=r}^{n-m}r!\left(\genfrac{}{}{0pt}{}{n}{m}\right)\left(\genfrac{}{}{0pt}{}{m}{j}\right)D_{m-j,\lambda }^{\left(r\right)}\left(x\right)C{h}_{j,\lambda }^{\left(r\right)}{S}_{2,\lambda }(l,r)S_1(n-m,l)=J_{n,\lambda }^{\left(r\right)}\left(x\right).\end{array}$$

(39)

Proof. 

From generating function $\left(1\right)$, we get

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& {\left(\frac{2log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\hfill \\ \hfill =& r!{\left(\frac{log(1+t)}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}-1}\right)}^r{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\frac{{({\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}-1)}^r}{r!}{\left(\frac{2}{{\left(1+\lambda log(1+t)\right)}^{\frac{1}{\lambda }}+1}\right)}^r\hfill \\ \hfill =& r!\sum _{n=0}^{\infty }{D}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}\sum _{l=r}^{\infty }{S}_{2,\lambda }(l,r)\frac{lo{g}^l(1+t)}{l!}\sum _{n=0}^{\infty }C{h}_{n,\lambda }^{\left(r\right)}\frac{t^n}{n!}\hfill \\ \hfill =& r!\sum _{n=0}^{\infty }\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)D_{n-j,\lambda }^{\left(r\right)}\left(x\right)C{h}_{j,\lambda }^{\left(r\right)}\frac{t^n}{n!}\sum _{l=r}^{\infty }{S}_{2,\lambda }(l,r)\sum _{n=l}^{\infty }{S}_1(n,l)\frac{t^n}{n!}\hfill \\ \hfill =& r!\sum _{n=0}^{\infty }\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)D_{n-j,\lambda }^{\left(r\right)}\left(x\right)C{h}_{j,\lambda }^{\left(r\right)}\frac{t^n}{n!}\sum _{n=r}^{\infty }\sum _{l=r}^n{S}_{2,\lambda }(l,r)S_1(n,l)\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{n=r}^{\infty }\sum _{m=0}^n\sum _{j=0}^m\sum _{l=r}^{n-m}r!\left(\genfrac{}{}{0pt}{}{n}{m}\right)\left(\genfrac{}{}{0pt}{}{m}{j}\right)D_{m-j,\lambda }^{\left(r\right)}\left(x\right)C{h}_{j,\lambda }^{\left(r\right)}{S}_{2,\lambda }(l,r)S_1(n-m,l)\frac{t^n}{n!}.\hfill \end{array}$$

Therefore, we obtain the required result. □

3. Identities about the Higher Order Degenerate Changhee–Genocchi Numbers and Polynomials of the Second Kind

In this section, by means of the Riordan matrix, we derive some new equalities between the higher order degenerate Changhee–Genocchi numbers and polynomials of the second kind and Striling numbers, Bell numbers, Changhee numbers, and Lah numbers.

Theorem 9.

For non-negative integer n, we have

$$\begin{array}{cc}\hfill & \sum _{k=0}^n\sum _{j=0}^{k}L(j,k)J_{k,\lambda }^{\left(m\right)}\left(x\right)S_2(n,k)={(-1)}^n{G}_{n,\lambda }^{\left(m\right)}\left(x\right).\hfill \end{array}$$

(40)

Proof. 

By Lemma 2 (19), we get

$$\begin{array}{c}\hfill R\left(\frac{k!}{n!}L(n,k)\right)=(1,\frac{-t}{1+t}).\end{array}$$

(41)

$$\begin{array}{cc}\hfill & \sum _{j=0}^{k}L(j,k)J_{k,\lambda }^{\left(m\right)}\left(x\right)=k!\sum _{j=0}^k\frac{j!L(j,k)}{k!}\frac{J_{k,\lambda }^{\left(m\right)}\left(x\right)}{j!}\hfill \\ \hfill =& k!\left[t^k\right]\left[{\left(\frac{2log(1+y)}{(1+\lambda log{(1+y)}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda log(1+y))}^{\frac{x}{\lambda }}\right|y=\frac{-t}{1+t}]\hfill \\ \hfill =& k!\left[t^k\right]{\left(\frac{-2log(1+t)}{(1-\lambda log{(1+t)}^{\frac{1}{\lambda }}+1}\right)}^m{(1-\lambda log(1+t))}^{\frac{x}{\lambda }}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \sum _{k=0}^n\sum _{j=0}^{k}L(j,k)J_{k,\lambda }^{\left(m\right)}\left(x\right)S_2(n,k)=n!\sum _{k=0}^n\frac{{\sum }_{j=0}^{k}L(j,k)J_{k,\lambda }^{\left(m\right)}}{k!}\frac{k!S_2(n,k)}{n!}\hfill \\ \hfill =& n!\left[t^n\right]\left[{\left(\frac{-2log(1+y)}{(1-\lambda log{(1+y)}^{\frac{1}{\lambda }}+1}\right)}^m{(1-\lambda log(1+y))}^{\frac{x}{\lambda }}\right|y=e^t-1]\hfill \\ \hfill =& n!\left[t^n\right]{\left(\frac{-2t}{{(1-\lambda t)}^{\frac{1}{\lambda }}+1}\right)}^m{(1-\lambda t)}^{\frac{x}{\lambda }}\hfill \\ \hfill =& n!\left[t^n\right]\sum _{n=0}^{\infty }{G}_{n,\lambda }^{\left(m\right)}\left(x\right){(-1)}^n\frac{t^n}{n!}={(-1)}^n{G}_{n,\lambda }^{\left(m\right)}\left(x\right).\hfill \end{array}$$

□

Corollary 10.

For $m=1$ in $\left(40\right)$, we have

$$\begin{array}{cc}\hfill & \sum _{k=0}^n\sum _{j=0}^{k}L(j,k)J_{k,\lambda }\left(x\right)S_2(n,k)={(-1)}^n{G}_{n,\lambda }\left(x\right).\hfill \end{array}$$

(42)

By means of $Lemma1$, the inverse relation $\left(16\right)$. This leads to the following conclusion,

Theorem 10.

For non-negative integer n, we have

$$\begin{array}{cc}\hfill & \sum _{k=0}^n{(-1)}^k{G}_{k,\lambda }^{\left(m\right)}\left(x\right)S_1(n,k)=\sum _{j=0}^{n}L(j,n)J_{j,\lambda }^{\left(m\right)}\left(x\right).\hfill \end{array}$$

(43)

Corollary 11.

For $m=1$ in $\left(43\right)$, we have

$$\begin{array}{cc}\hfill & \sum _{k=0}^n{(-1)}^k{G}_{k,\lambda }\left(x\right)S_1(n,k)=\sum _{j=0}^{n}L(j,n)J_{j,\lambda }\left(x\right).\hfill \end{array}$$

(44)

Theorem 11.

For non-negative integer n, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n{S}_2(n,k;r)J_{k,\lambda }^{\left(m\right)}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)r^{n-l}{G}_{l,\lambda }^{\left(m\right)}\left(x\right).\end{array}$$

(45)

Proof. 

By using $\left(19\right)$, we see that

$$\begin{array}{c}\hfill R\left(\frac{k!}{n!}{S}_2(n,k;r)\right)=(e^{rt},e^t-1).\end{array}$$

(46)

$$\begin{array}{cc}\hfill & \sum _{k=0}^n{S}_2(n,k;r)J_{k,\lambda }^{\left(m\right)}\left(x\right)=n!\sum _{k=0}^n\frac{k!S_2(n,k;r)}{n!}\frac{J_{k,\lambda }^{\left(m\right)}\left(x\right)}{k!}\hfill \\ \hfill =& n!\left[t^n\right]e^{rt}\left[{\left(\frac{2log(1+y)}{(1+\lambda log{(1+y)}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda log(1+y))}^{\frac{x}{\lambda }}\right|y=e^t-1]\hfill \\ \hfill =& n!\left[t^n\right]e^{rt}{\left(\frac{2t}{{(1+\lambda t)}^{\frac{1}{\lambda }}}\right)}^m{(1+\lambda t)}^{\frac{x}{\lambda }}=n!\left[t^n\right]e^{rt}\sum _{n=0}^{\infty }{G}_{n,\lambda }^{\left(m\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& n!\left[t^n\right]\sum _{n=0}^{\infty }{r}^n\frac{t^n}{n!}\sum _{n=0}^{\infty }{G}_{n,\lambda }^{\left(m\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)r^{n-l}{G}_{l,\lambda }^{\left(m\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)r^{n-l}{G}_{l,\lambda }^{\left(m\right)}\left(x\right).\hfill \end{array}$$

Therefore, we obtain the required result. □

Corollary 12.

For $m=1$ in $\left(45\right)$, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n{S}_2(n,k;r)J_{k,\lambda }\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)r^{n-l}{G}_{l,\lambda }\left(x\right).\end{array}$$

(47)

Corollary 13.

For $x=0$ in $\left(47\right)$, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n{S}_2(n,k;r)J_{k,\lambda }=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)r^{n-l}{G}_{l,\lambda }.\end{array}$$

(48)

Corollary 14.

For $r=0$ in $\left(47\right)$, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n{S}_2(n,k)J_{k,\lambda }\left(x\right)=G_{l,\lambda }\left(x\right).\end{array}$$

(49)

By the inverse relation $\left(17\right)$, we have

Theorem 12.

For non-negative integer n, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n{S}_1(n,k;r)G_{k,\lambda }^{\left(m\right)}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right){(-r)}_{n-l}{J}_{l,\lambda }^{\left(m\right)}\left(x\right).\end{array}$$

(50)

Which completes the proof.

Theorem 13.

For integer $n\ge 1$, we have

$$\begin{array}{c}\hfill \sum _{k=1}^n\sum _{j=1}^{k}B(k,j)J_{j,\lambda }^{\left(m\right)}\left(x\right)S_1(n,k)=G_{n,\lambda }^{\left(m\right)}\left(x\right).\end{array}$$

(51)

Proof. 

By $\left(19\right)$, we get

$$\begin{array}{c}\hfill R\left(\frac{k!}{n!}B(n,k)\right)=(1,e^{e^t-1}-1).R\left(\frac{k!}{n!}{S}_1(n,k)\right)=(1,log(1+t)).\end{array}$$

(52)

$$\begin{array}{cc}\hfill & \sum _{j=1}^{k}B(k,j)J_{j,\lambda }^{\left(m\right)}\left(x\right)=k!\sum _{j=1}^k\frac{j!B(k,j)}{k!}\frac{J_{j,\lambda }^{\left(m\right)}\left(x\right)}{j!}\hfill \\ \hfill =& k!\left[t^k\right]\left[{\left(\frac{2log(1+y)}{(1+\lambda log{(1+y)}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda log(1+y))}^{\frac{x}{\lambda }}\right|y=e^{e^t-1}-1]\hfill \\ \hfill =& k!\left[t^k\right]{\left(\frac{2\left(e^t-1\right)}{(1+\lambda {(e^t-1)}^{\frac{1}{\lambda }}+1}\right)}^m{(e^t-1)}^{\frac{x}{\lambda }}.\hfill \\ \hfill & \sum _{k=1}^n\sum _{j=1}^{k}B(k,j)J_{j,\lambda }^{\left(m\right)}\left(x\right)S_1(n,k)=n!\sum _{k=1}^n\frac{{\sum }_{j=1}^{k}B(k,j)J_{j,\lambda }^{\left(m\right)}\left(x\right)}{k!}\frac{k!S_1(n,k)}{n!}\hfill \\ \hfill =& n!\left[t^n\right]\left[{\left(\frac{2\left(e^y-1\right)}{(1+\lambda {(e^y-1)}^{\frac{1}{\lambda }}+1}\right)}^m{(e^y-1)}^{\frac{x}{\lambda }}\right|y=log(1+t)]\hfill \\ \hfill =& n!\left[t^n\right]{\left(\frac{2t}{{(1+\lambda t)}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda t)}^{\frac{x}{\lambda }}=n!\left[t^n\right]\sum _{n=0}^{\infty }{G}_{n,\lambda }^{\left(m\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& G_{n,\lambda }^{\left(m\right)}\left(x\right).\hfill \end{array}$$

We acquire the desired result. □

Corollary 15.

For $m=1$ in $\left(51\right)$, we have

$$\begin{array}{c}\hfill \sum _{k=1}^n\sum _{j=1}^{k}B(k,j)J_{j,\lambda }\left(x\right)S_1(n,k)=G_{n,\lambda }\left(x\right).\end{array}$$

(53)

By the inverse relation $\left(16\right)$, we have

Theorem 14.

For integer $n\ge 1$, we have

$$\begin{array}{c}\hfill \sum _{k=1}^n{G}_{k,\lambda }^{\left(m\right)}\left(x\right)S_2(n,k)=\sum _{j=1}^{n}B(n,j)J_{j,\lambda }^{\left(m\right)}\left(x\right).\end{array}$$

(54)

Theorem 15.

For integer $n\ge 1$, we have

$$\begin{array}{c}\hfill \sum _{k=0}^n{(-2)}^{n-k}\left(\genfrac{}{}{0pt}{}{n}{k}\right)C{h}_{n-k}^{\left(r\right)}{J}_{k,\lambda }^{\left(m\right)}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{r+l-1}{l}\right){\left(n\right)}_l{J}_{n-l,\lambda }^{\left(m\right)}\left(x\right).\end{array}$$

(55)

Proof. 

From $\left(19\right)$, we note that

$$\begin{array}{cc}\hfill & R\left({(-2)}^{n-k}\frac{C{h}_{n-k}^{\left(r\right)}}{(n-k)!}\right)=(\frac{1}{{(1-t)}^r},t).\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & \sum _{k=0}^n{(-2)}^{n-k}\left(\genfrac{}{}{0pt}{}{n}{k}\right)C{h}_{n-k}^{\left(r\right)}{J}_{k,\lambda }^{\left(m\right)}\left(x\right)\hfill \\ \hfill =& n!\left[t^n\right]\frac{1}{{(1-t)}^r}\left[{\left(\frac{2log(1+y)}{(1+\lambda log{(1+y)}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda log(1+y))}^{\frac{x}{\lambda }}\right|y=t]\hfill \\ \hfill =& n!\left[t^n\right]\frac{1}{{(1-t)}^r}{\left(\frac{2log(1+t)}{(1+\lambda log{(1+t)}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda log(1+t))}^{\frac{x}{\lambda }}\hfill \\ \hfill =& n!\left[t^n\right]\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(m\right)}\left(x\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }\left(\genfrac{}{}{0pt}{}{r+n-1}{n}\right)t^n\hfill \\ \hfill =& n!\left[t^n\right]\sum _{n=0}^{\infty }\sum _{l=0}^n{J}_{n-l,\lambda }^{\left(m\right)}\left(x\right)\frac{1}{(n-l)!}\left(\genfrac{}{}{0pt}{}{r+l-1}{l}\right)t^n\hfill \\ \hfill =& \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{r+l-1}{l}\right){\left(n\right)}_l{J}_{n-l,\lambda }^{\left(m\right)}\left(x\right).\hfill \end{array}$$

Which completes the proof. □

Theorem 16.

For integer $n\ge 1$, we have

$$\begin{array}{cc}\hfill & \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right){\alpha }^{n-l}{\beta }^{l}C{h}_l^{\left(k\right)}{J}_{n-l,\lambda }^{\left(m\right)}\left(x\right)=\sum _{j=1}^n\frac{C{G}_n^{k,j}(\alpha ,\beta )G_{j,\lambda }^{\left(m\right)}\left(x\right)}{j!}.\hfill \end{array}$$

(57)

Proof. 

By $\left(19\right)$, we get

$$\begin{array}{cc}\hfill & R\left(\frac{C{G}_n^{k,j}(\alpha ,\beta )}{n!}\right)=(2^k{(2+\beta t)}^{-k},log(1+\alpha t)).\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill & \sum _{j=1}^n\frac{C{G}_n^{k,j}(\alpha ,\beta )}{n!}\frac{G_{j,\lambda }^{\left(m\right)}\left(x\right)}{j!}\hfill \\ \hfill =& \left[t^n\right]2^k{(2+\beta t)}^{-k}\left[{\left(\frac{2y}{{(1+\lambda y)}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda y)}^{\frac{x}{\lambda }}\right|y=log(1+\alpha t)]\hfill \\ \hfill =& \left[t^n\right]{\left(\frac{2}{2+\beta t}\right)}^k{\left(\frac{2log(1+\alpha t)}{{(1+\lambda log(1+\alpha t))}^{\frac{1}{\lambda }}+1}\right)}^m{(1+\lambda log(1+\alpha t))}^{\frac{x}{\lambda }}\hfill \\ \hfill =& \left[t^n\right]\sum _{l=0}^{\infty }C{h}_l^{\left(k\right)}{\beta }^l\frac{t^l}{l!}\sum _{n=0}^{\infty }{J}_{n,\lambda }^{\left(m\right)}\left(x\right){\alpha }^n\frac{t^n}{n!}\hfill \\ \hfill =& \left[t^n\right]\sum _{n=0}^{\infty }\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right){\alpha }^{n-l}{\beta }^{l}C{h}_l^{\left(k\right)}{J}_{n-l,\lambda }^{\left(m\right)}\left(x\right)\frac{t^n}{n!}\hfill \\ \hfill =& \sum _{l=0}^n\frac{1}{(n-l)!l!}{\alpha }^{n-l}{\beta }^{l}C{h}_l^{\left(k\right)}{J}_{n-l,\lambda }^{\left(m\right)}\left(x\right).\hfill \end{array}$$

Which completes the proof. □

Corollary 16.

For $\alpha =\beta$ in $\left(57\right)$, we have

$$\begin{array}{cc}\hfill & \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right){\alpha }^{n}C{h}_l^{\left(k\right)}{J}_{n-l,\lambda }^{\left(m\right)}\left(x\right)=\sum _{j=1}^n\frac{C{G}_n^{k,j}(\alpha ,\alpha )G_{j,\lambda }^{\left(m\right)}\left(x\right)}{j!}.\hfill \end{array}$$

(59)

Corollary 17.

For $m=1$ in $\left(57\right)$, we have

$$\begin{array}{cc}\hfill & \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right){\alpha }^{n-l}{\beta }^{l}C{h}_l^{\left(k\right)}{J}_{n-l,\lambda }\left(x\right)=\sum _{j=1}^n\frac{C{G}_n^{k,j}(\alpha ,\beta )G_{j,\lambda }\left(x\right)}{j!}.\hfill \end{array}$$

(60)

4. Conclusions

The degenerate Changhee–Genocchi numbers and polynomials of the second kind are a combinatorial sequence that have been generalized to Changhee–Genocchi polynomials in recent years. In this paper, the basic properties of higher order degenerate Changhee–Genocchi numbers and polynomials of the second kind are given by using generating functions and the Riordan matrix methods. Additionally, the combinatorial constants between the higher order degenerate Changhee–Genocchi numbers and polynomials of the second kind and other special combinatorial sequences are derived. Next, we can further study and generalize them using probabilistic methods, asymptotic methods, etc. More importantly, degenerate Changhee–Genocchi numbers and polynomials of the second kind can be used not only in mathematics, mathematical physics equations, computer science, etc.