Properties of the Poly-Changhee Polynomials and Its Applications

Abstract

The introduction of Changhee polynomials and numbers by Kim et al. in 2013, as published in Advances in Studies in Theoretical Physics, Volume 7, 2013, No. 20, pages 993–1003, sparked interest in their properties and identities. Building on Kim’s work, the authors of this chapter define the generating functions for poly-Changhee polynomials and numbers, as well as higher-order poly-Changhee polynomials and numbers, using an umbral calculus approach. They demonstrate how these polynomials and numbers are closely related to other polynomials, such as Stirling polynomials of the first and second kind, and Daehee polynomials and numbers, by utilizing poly-logarithmic functions. The authors also derive new explicit formulas and identities for these polynomials and numbers in their research.

1. Introduction

In recent years, there has been significant research on Changhee polynomials and their properties, generalizations, and applications in various fields. Here is a brief overview of some of the notable works in this area:

• Kim et al. [1]: Introduced Changhee polynomials and numbers using an umbral calculus approach and obtained interesting identities and properties of these polynomials.

• Kim et al. [2]: Considered Witt-type formulas for Changhee numbers and polynomials, which are a type of recurrence relation satisfied by these numbers and polynomials.

• Kim and Kim [3]: Introduced higher-order Changhee polynomials and established relations between higher-order Changhee polynomials and special polynomials.

• Rim et al. [4]: Considered Witt-type formulas for $n$-th twisted Changhee numbers and polynomials, which are a generalization of Changhee numbers and polynomials with a twist parameter $n$.

• Jang et al. [5]: Investigated higher-order twisted Changhee polynomials and numbers, and discussed computations of zeros of these polynomials.

• Kim et al. [6, 7]: Established nonlinear differential equations satisfied by Changhee polynomials, which turned out to be useful for studying special polynomials and mathematical physics.

Moreover, Changhee polynomials and numbers have found applications in various fields, including mathematics, mathematical physics, computer science, engineering sciences, and real-world problems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. These polynomials and numbers have been used to solve problems in diverse areas, showcasing their versatility and importance in different disciplines.

Additionally, the notation $\mathbb{N}$ denotes the set of natural numbers, and ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ denotes the set of non-negative integers.

For $t\in {\mathbb{C}}_p$ with ${\left|t\right|}_p<p^{-\frac{1}{p-1}}$, the Changhee polynomials are defined by the generating function to be

$$\frac{2}{t+2}{\left(1+t\right)}^x=\sum _{n=0}^{\infty }{\mathit{Ch}}_n\left(x\right)\frac{t^n}{n!}\left(\mathrm{see}\left[20,23,25,30\right]\right).$$E1

For $x=0,{\mathit{Ch}}_n={\mathit{Ch}}_n\left(0\right)=\frac{n!}{2^n}$ are called Changhee numbers [1]. The first few Changhee numbers ${\mathit{Ch}}_n$ are ${\mathit{Ch}}_0=1,{\mathit{Ch}}_1=\frac{1}{2},{\mathit{Ch}}_2=\frac{1}{2},{\mathit{Ch}}_3=\frac{3}{4},{\mathit{Ch}}_4=\frac{3}{2},{\mathit{Ch}}_5=\frac{15}{4},\dots$.

From (1), we have

$${\mathit{Ch}}_n\left(x\right)=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\left(x\right)}_{n-l}{\mathit{Ch}}_l,$$E2

where ${\left(x\right)}_n$ is a falling factorial (also called the Pochhammer symbol and related to the gamma function in such a way that $\frac{\mathrm{\Gamma }\left(x+n\right)}{\mathrm{\Gamma }\left(x\right)},$ for detail (see [11, 17, 26, 41]).

$${\left(x\right)}_n=x\left(x-1\right)\left(x-2\right)\cdots \left(x-n+1\right)\left(n\in \mathbb{N}\cup \left\{0\right\}\right).$$E3

Further, to find the remarkable results in the following section, we need to define classical Bernoulli $B_n\left(x\right)$ [16, 21, 33], Bernoulli polynomials of the second kind $b_n\left(x\right)$ [14], modified poly-Bernoulli numbers $C_n^{\left(\mathrm{k}\right)}$ [32], poly-Daehee numbers $D_n^{\left(k\right)}\left(x\right)$ [15] by means of the generating functions, so before defining all these definitions, first we need to define generating function in a nutshell.

2. Generating functions

The generating function is a way of encoding an infinite sequence of numbers $⟨a_0,a_1,a_2\cdots ⟩$ by treating them as a coefficient of a formal power series. Then, the series is called the generating function of the series. For example, if $⟨a⟩=⟨a_0,a_1,a_2\cdots ⟩$ be a sequence of terms, then generating function (in short we call it GF) of the above sequence is an infinite series $G\left(x\right)={\sum }_{n=0}^{\infty }{a}_n{x}^n$.

• If $⟨a_n⟩=c\left(c\ne 0\right)$, then GF becomes $G\left(x\right)=\frac{c}{1-x}$.

• If $⟨a_n⟩=\left(\begin{array}{c}\left(\begin{array}{c}n\\ k\end{array}\right),k\le n\\ 0,k>n\end{array}\right.$, then GF becomes $G\left(x\right)={\left(1+x\right)}^n$.

• If $⟨a_n⟩=c\left(c\ne 0\right)$, then GF becomes $G\left(x\right)=\frac{c}{1-x}$.

• If $⟨a_n⟩=\frac{n!}{2^n}$, then GF becomes $G\left(x\right)=\frac{2}{2+x}={\mathit{Ch}}_n\left(0\right)$, the Changhee numbers described in first section. Similarly, if we choose $⟨a_n⟩={\left(\frac{1}{2}\right)}^n$, then we also get Changhee numbers. So, here we have seen that the generating function has an elementary form. In fact, more simple than the sequence itself, this is the first magic of the generating function; in many natural instances, the generating function turns out to be very simple.

There are various generating functions, such as ordinary, exponential, Bell series, etc. Similarly, we can find generating function in $x$ and $t$, denoted by $G\left(x,t\right)$ and defined by the formal power series as

$$G\left(x,t\right)=\sum _{n=0}^{\infty }{g}_n\left(x\right)t^n,$$E4

and we say that $G\left(x,t\right)$ has generted the set $g_n\left(x\right)$. We can extend the above definition slightly by the form

$$G\left(x,t\right)=\sum _{n=0}^{\infty }{c}_n{g}_n\left(x\right)t^n,$$E5

where $c_n$ be sequence independent of $x$ and $t$. Certain properties of the polynomial set $g_n\left(x\right)$ are readily deduced from the known properties of $G\left(x,t\right)$. In this decades, many of researchers found the generating function of various types of polynomial and also generalize the idea to get new set of polynomial and numbers. In this regard, many authors can visit the references. Also, generating function plays a large role in our study of the polynomial sets. For example,

• The Legender polynomials (denoted by $P_n\left(x\right)$) is defined by the GF

$$G\left(x,t\right)=\frac{1}{\sqrt{\left(1-2\mathit{xt}+t^2\right)}}=\sum _{n=0}^{\infty }{P}_n\left(x\right)t^n.$$

• The Hermite polynomial (denoted by $H_n\left(x\right)$) is defined by the GF

$$G\left(x,t\right)=\mathit{exp}\left(2\mathit{xt}-t^2\right)=\sum _{n=0}^{\infty }{H}_n\left(x\right)t^n.$$

Note 2.1. Let $\mathcal{A}\left(x\right)$ and $\mathcal{B}\left(x\right)$ are two GF of the two sequences $⟨a_n⟩$ and $⟨b_n⟩$, then their product $\mathcal{A}\left(x\right)$. $\mathcal{B}\left(x\right)$ is also a GF.

Now, here, the generating function of Bernoulli plynomial of first kind, second kind, (for) is given by $G\left(x,t\right)=\frac{t{e}^{\mathit{xt}}}{e^t-1}={\sum }_{n=0}^{\infty }\frac{B_n\left(x\right)}{n!}{t}^n$ and $G\left(x,t\right)=\frac{{\left(1+t\right)}^x}{log\left(1+t\right)}t={\sum }_{n=0}^{\infty }\frac{b_n\left(x\right)}{n!}{t}^n$, these comes under the category of exponential generating functions.

The GF of modified poly-Bernoulli numbers $C_n^{\left(\mathrm{k}\right)}$ (see [32]), poly-Daehee numbers $D_n^{\left(k\right)}\left(x\right)$ (see [15]) are given as:

$$\frac{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}{e^t-1}=\sum _{n=0}^{\infty }\frac{C_n^{\left(\mathrm{k}\right)}}{n!}{t}^n,\left(|t|<\pi ,\mathrm{k}\in {\mathbb{Z}}^{+}\right)$$E6

$$\frac{log\left(1+t\right)}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}=\sum _{n=0}^{\infty }\frac{D_n^{\left(\mathrm{k}\right)}}{n!}{t}^n,\left(|t|<\pi ,\mathrm{k}\in {\mathbb{Z}}^{+}\right).$$E7

As we know that the Stirling numbers are the coefficients of the expansion of the falling and rising factorial, denoted by ${\left(x\right)}_n,{\left(x\right)}^n$, respectively. Falling and rising factorials are polynomials in degree $n$. The Stirling numbers of first kind $S_1\left(n,k\right)$ and the Stirling number of second kind $S_2\left(n,k\right)$ are generated by the following series expansion

$$\frac{{\left(log\left(1+t\right)\right)}^k}{k!}=\sum _{n=0}^{\infty }{S}_1\left(n,k\right)\frac{t^n}{n!}\text{and}\frac{{\left(e^t-1\right)}^k}{k!}=\sum _{n=0}^{\infty }{S}_2\left(n,k\right)\frac{t^n}{n!}.$$E8

In view of Eq. (6), we can also obtain the following another connections of Stirling numbers.

$$x^{\left(n\right)}=\sum _{k=0}^n{S}_2\left(n,k\right){\left(x\right)}_k,$$E9

where $S_2\left(n,0\right)={\delta }_{n,0},S_2\left(n,k\right)=0,$ for $k>n,$ and ${\delta }_{n,k}$ is the kronecker delta.

The Stirling number of the first kind, denoted by $S_1\left(n,k\right)$ is given as:

$${\left(x\right)}_n=\sum _{l=0}^n{S}_1\left(n,l\right)x^l,\left(n\ge 0\right)$$E10

${\left(x\right)}_n$ is called falling factorial, given by Eq. (3). After expanding falling factorial and right hand side of the above equation, we get list of Stirling numbers for different values of $n$.

Remark 2.1. One of these falling and raising factorials can also be found by using

$${\left(x\right)}^n={\left(-1\right)}^n{\left(-x\right)}_n$$

For $\mathrm{k}\in \mathbb{Z},\left(\mathrm{k}>1\right)$, the classical polylogarithm function ${\mathrm{Li}}_{\mathrm{k}}\left(x\right)$ (see [2, 30, 34]) is defined by

$${\mathrm{Li}}_{\mathrm{k}}\left(x\right)=\sum _{n=1}^{\infty }\frac{x^n}{n^{\mathrm{k}}},\left(|x|<1\right).$$E11

Note 2.2. For $k=1,{\mathrm{Li}}_1\left(x\right)={\sum }_{n=1}^{\infty }\frac{x^n}{n}=\frac{1}{log\left(1-x\right)}$, (here the sequuence $⟨a_n⟩=\frac{\left(n-1\right)!}{n!}).$.

In this chapter, we consider the poly-Changhee numbers and polynomials and derive new explicit formulas and identities for those numbers and polynomials.

3. Higher-order Changhee polynomials and numbers

As one of the special polynomial, the Changhee polynomials and numbers are closely related to some other important polynomials and numbers such as combinatorics, applied science, mathematical physics, etc. The generating function of the higher-order Changhee polynomials is described as:

$${\left(\frac{2}{t+2}\right)}^k{\left(1+t\right)}^x=\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(k\right)}\left(x\right)\frac{t^n}{n!}.$$E12

where the sequence of the polynomial ${\mathit{Ch}}_n^{\left(k\right)}\left(x\right)$ is called a higher-order Changhee polynomial. For the different values of $k,\left(k\in \mathbb{N}\right)$ and at $x=0$, the constant terms of the Changhee polynomials are listed here:

$$\begin{array}{c}{\mathit{Ch}}_n^{\left(0\right)}\left(0\right)=1\\ {\mathit{Ch}}_n^{\left(1\right)}\left(0\right)=\frac{k}{2}\\ {\mathit{Ch}}_n^{\left(2\right)}\left(0\right)=\frac{k^2-k}{4}\\ {\mathit{Ch}}_n^{\left(3\right)}\left(0\right)=\frac{k^3-3k^2+4k}{8}\\ {\mathit{Ch}}_n^{\left(3\right)}\left(0\right)=\frac{k^4-4k^3+19k^2-28k}{16}\\ \cdots \\ \cdots \end{array}$$

Note 3.1. For $k=1$ in Eq. (10), we get simple Changhee polynomials, given by the Eq. (1).

Similarly, we can find other constant terms of the above set of Changhee Polynomial given by (10). Now, in general, some of the higher-order Changhee polynomials are also listed here (the generating function is given in (10)):

$$\begin{array}{c}{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)=1\\ {\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)=x+\frac{k}{2}\\ {\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)=x^2+\left(k-1\right)x+\frac{k^2-k}{4}\\ {\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)=x^3+\left(\frac{k}{2}-1\right)x^2+\left(\frac{8k^2-9k+8}{4}\right)x+\left(\frac{k^3-3k^2+4k}{8}\right)\\ {\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)=x^4+\left(2k-6\right)x^3+\left(\frac{3k^2-12k+22}{2}\right)x^2+\left(\frac{k^3-3k^2+4k}{2}\right)x+\left(\frac{k^4-6k^3+19k^2-28k}{6}\right)\\ \cdots \\ \cdots \end{array}$$

Note 3.2. For different choices of $k$ and $n$, we can easily find various simple Changhee polynomials and higher-order Changhee polynomials and numbers.

In the next section, we shall define a simple poly-Changhee polynomial using the GF of the Changhee polynomial and the GF of the poly-logarithmic function.

4. Poly-Changhee polynomials and numbers

For $\mathrm{k}\in \mathbb{Z}$, the poly-Changhee polynomials are defined by the following generating function to be

$$\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}=\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!},\left(n\in {\mathbb{N}}_0\right).$$E13

For $x=0,{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}={\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(0\right)$ are called poly-Changhee numbers, given by

$$\frac{2t}{\left(2+t\right){\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}=\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(0\right)\frac{t^n}{n!},$$E14

and by (12) we can easily obtain ${\mathit{Ch}}_0^{\left(\mathrm{k}\right)}=0$.

For the case $\mathrm{k}=1$, we have

$$\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_1\left(1-e^{-t}\right)}=\frac{2{\left(1+t\right)}^x}{2+t}=\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(1\right)}\left(x\right)\frac{t^n}{n!}.$$E15

From (12) and (13), we get

$${\mathit{Ch}}_n^{\left(1\right)}\left(x\right)={\mathit{Ch}}_n\left(x\right),\left(n\ge 0\right).$$E16

Theorem 4.1. Let $n\in {\mathbb{Z}}^{\ast }=\left\{0\right\}\cup {\mathbb{Z}}^{+}$, where ${\mathbb{Z}}^{+}$ denotes the positive integer. Then we have

$$\begin{array}{c}{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)=\sum _{l=0}^n{\mathit{Ch}}_l^{\left(\mathrm{k}\right)}{\left(x\right)}_{n-l}\\ =\sum _{l=0}^n\left(n-l\right)!\left(\begin{array}{c}n\\ l\end{array}\right)\left(\begin{array}{c}x\\ n-l\end{array}\right){\mathit{Ch}}_l^{\left(\mathrm{k}\right)}.\end{array}$$E17

Proof. Now from (11) the generating function of poly-Changhee polynomials, we have

$$\begin{array}{l}\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!}=\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\\ =\left(\frac{2t}{\left(2+t\right){\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\right){\left(1+t\right)}^x\\ =\left(\sum _{l=0}^{\infty }\frac{{\mathit{Ch}}_l^{\left(\mathrm{k}\right)}\left(0\right)t^l}{l!}\right)\left(\sum _{n=0}^{\infty }\frac{{\left(x\right)}_n{t}^n}{n!}\right)\\ =\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }\frac{{\mathit{Ch}}_l^{\left(\mathrm{k}\right)}\left(0\right){\left(x\right)}_n{t}^{n+l}}{n!l!}\\ =\sum _{n=0}^{\infty }\sum _{l=0}^n\frac{{\mathit{Ch}}_l^{\left(\mathrm{k}\right)}\left(0\right){\left(x\right)}_{n-l}{t}^n}{\left(n-l\right)!l!}\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(n-l\right)!\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_l^{\left(\mathrm{k}\right)}\left(\begin{array}{c}x\\ n-l\end{array}\right)\right)\frac{t^n}{n!}.\end{array}$$E18

Thus, we obtain the above result on equating the coefficients of $t^n$ on both sides of Eq. (16).

Theorem 4.2. Let $n\in {\mathbb{Z}}^{\ast }=\left\{0\right\}\cup {\mathbb{Z}}^{+}$, where ${\mathbb{Z}}^{+}$ denotes the positive integer. Then we have

$${\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)=\frac{{\mathit{Ch}}_{n+1}^{\left(\mathrm{k}\right)}\left(x+1\right)-{\mathit{Ch}}_{n+1}^{\left(\mathrm{k}\right)}\left(x\right)}{n+1}.$$E19

Proof. Again from (11) the generating function of poly-Changhee polynomials, we have

$$\begin{array}{l}\sum _{n=0}^{\infty }\left({\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x+1\right)-{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)\right)\frac{t^n}{n!}\\ =\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^{x+1}}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}-\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\\ =t\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\\ =\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^{n+1}}{n!}.\end{array}$$E20

Thus, we acquire the proof of the theorem on equating the coefficients of $t^{n+1}$ on both sides of Eq. (18).

Theorem 4.3. Let $n\in {\mathbb{Z}}^{\ast }=\left\{0\right\}\cup {\mathbb{Z}}^{+}$, where ${\mathbb{Z}}^{+}$ denotes the positive integer. Then we have

$${\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x+y\right)=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\left(y\right)}_l.$$E21

Proof. From (11) the generating function of poly-Changhee polynomials, we have

$$\begin{array}{l}\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x+y\right)\frac{t^n}{n!}=\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^{x+y}}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\\ =\left(\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\right){\left(1+t\right)}^y\\ =\left(\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!}\right)\left(\sum _{l=0}^{\infty }\frac{{\left(y\right)}_l{t}^l}{l!}\right)\\ =\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }\frac{{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right){\left(y\right)}_l{t}^{n+l}}{l!n!}\\ =\sum _{n=0}^{\infty }\sum _{l=0}^n\frac{{\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\left(y\right)}_l{t}^n}{l!\left(n-l\right)!}\\ =\sum _{n=0}^{\infty }\sum _{l=0}^n\left(\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\left(y\right)}_l\right)\frac{t^n}{n!}\end{array}$$E22

Thus, we acquire the above theorem on equating the coefficient of $t^n$ on both sides of Eq. (20).

Theorem 4.4. For $n\ge 0$,

$$\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\mathrm{C}}_l^{\left(k\right)}=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_{n-l}^k\left(x\right){\mathcal{B}}_l$$

For every integer $k$ poly-Bernoulli number ${\mathfrak{B}}_{\mathfrak{n}}^{\left(\mathfrak{k}\right)}$ and the modified poly-Bernoulli numbers ${\mathrm{C}}_{\mathrm{n}}^{\left(\mathrm{k}\right)}$ introduced by Kaneko [23].

$$\frac{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}{1-e^{-t}}=\sum _{n=0}^{\infty }\frac{{\mathfrak{B}}_{\mathfrak{n}}^{\left(\mathfrak{k}\right)}}{n!}{t}^n$$E23

$$\frac{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}{e^t-1}=\sum _{n=0}^{\infty }\frac{{\mathrm{C}}_{\mathrm{n}}^{\left(\mathrm{k}\right)}}{n!}{t}^n$$E24

Now, from the definition of poly-changhee polynomials, we observe that

$$\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{e^t-1}=\left(\frac{\left(2t\right){\left(1+t\right)}^x}{\left(2+t\right)\left({\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)\right)}\right)\frac{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}{e^t-1}$$E25

$$\begin{array}{l}=\left(\sum _{n=0}^{\infty }\frac{{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)}{n!}{t}^n\right)\left(\sum _{l=0}^{\infty }\frac{{\mathrm{C}}_l^{\left(k\right)}}{l!}{t}^l\right)\\ =\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }\frac{{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right){\mathrm{C}}_l^{{\left(k\right)}_l}{t}^{n+l}}{l!n!}\\ =\sum _{n=0}^{\infty }\sum _{l=0}^n\frac{{\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\mathrm{C}}_l^{\left(k\right)}}{l!\left(n-l\right)!}\frac{t^n}{n!}\left(n!\right)\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\mathrm{C}}_l^{\left(k\right)}\right)\frac{t^n}{n!}\end{array}$$E26

Now, left-hand side of the Eq. (24), we have

$$\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{e^t-1}=\left(\frac{2{\left(1+t\right)}^x}{2+t}\right)\left(\frac{t}{e^t-1}\right)$$E27

$$\begin{array}{l}=\left(\sum _{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!}\right)\left(\sum _{l=0}^{\infty }{\mathcal{B}}_l\frac{t^l}{l!}\right)\\ =\sum _{n=0}^{\infty }\sum _{l=0}^n\frac{{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right){\mathcal{B}}_l}{\left(n-l\right)!l!}{t}^{n+l}\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^{n}n\begin{array}{c}\\ l\end{array}{\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\mathcal{B}}_l\right)\frac{t^n}{n!}\end{array}$$E28

From Eqs. (25) and (27), we have

$$\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_{n-l}^{\left(\mathrm{k}\right)}\left(x\right){\mathrm{C}}_l^{\left(k\right)}=\sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right){\mathit{Ch}}_{n-l}^k\left(x\right){\mathcal{B}}_l$$E29

Therefore, we arrive the above result (Table 1).

S. No.	Name of Polynomial	$\mathcal{A}\left(y\right)$	Generating Function
I.	Changhee polynomial [13]	$\left(\frac{2}{2+t}\right){\left(1+t\right)}^x$	$\begin{array}{l}{\sum }_{n=0}^{\infty }\frac{t^n}{n!}{\mathit{Ch}}_n\left(x\right)\\ =\frac{2}{2+t}{\left(1+t\right)}^x\end{array}$
II.	Higher-order Changhee polynomial [3]	${\left(\frac{2}{2+t}\right)}^k{\left(1+t\right)}^x$	$\begin{array}{l}{\sum }_{n=0}^{\infty }\frac{t^n}{n!}{\mathit{Ch}}_n^{\left(k\right)}\left(x\right)\\ ={\left(\frac{2}{2+t}\right)}^k{\left(1+t\right)}^x\end{array}$
III.	Degenerate Changhee polynomial [38]	$\left(\frac{2\lambda }{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right){\left(1+log{\left(1+\mathit{\lambda t}\right)}^{\frac{1}{\lambda }}\right)}^x$	$\begin{array}{l}{\sum }_{n=0}^{\infty }\frac{t^n}{n!}{\mathit{Ch}}_n^{\ast }\left(x\right)\\ =\left(\frac{2\lambda }{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right){\left(1+log\left(1+\mathit{\lambda t}\right)\right)}^{\frac{x}{\lambda }}\end{array}$
IV.	Higher-order degenerate Changhee polynomial [35]	${\left(\frac{2\lambda }{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right)}^{\left(k\right)}{\left(1+log{\left(1+\mathit{\lambda t}\right)}^{\frac{1}{\lambda }}\right)}^x$	$\begin{array}{l}{\sum }_{n=0}^{\infty }\frac{t^n}{n!}{\mathit{Ch}}_n^{\ast ,\left(k\right)}\left(x\right)\\ =\left(\frac{2\lambda }{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right){\left(1+log\left(1+\mathit{\lambda t}\right)\right)}^{\frac{x}{\lambda }}\end{array}$
V.	Poly-Changhee polynomial	$\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}$	$\begin{array}{l}\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\\ ={\sum }_{n=0}^{\infty }{\mathit{Ch}}_n^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!}\end{array}$

Table 1.

Similar members to the Changhee family ${\mathit{Ch}}_n\left(x\right)$.

5. Conclusion

In this chapter, we have studied Changhee polynomial and their various generalization. In the same way, we can also define degenerate poly-Changhee polynomial by the following generating function. For $\mathrm{k}\in \mathbb{Z}$, the degenerate poly-Changhee polynomials are defined by the following generating function:

$$\left(\frac{2\mathit{\lambda t}}{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right)\frac{{\left(1+log{\left(1+\mathit{\lambda t}\right)}^{\frac{1}{\lambda }}\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}=\sum _{n=0}^{\infty }{\mathit{Ch}}_{n,\lambda }^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!},\left(n\in {\mathbb{N}}_0\right).$$E30

Remark 5.1. The limiting case (i.e., when ${lim}_{\lambda \to 0}$) of degenerate poly-Changhee polynomials gives poly-Changhee polynomials (for more details, visit [12, 31, 33, 36]).

$$\begin{array}{c}\underset{\lambda \to 0}{lim}\left\{\sum _{n=0}^{\infty }{\mathit{Ch}}_{n,\lambda }^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!}\right\}=\underset{\lambda \to 0}{lim}\left\{\left(\frac{2\mathit{\lambda t}}{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right)\frac{{\left(1+log\left(1+\mathit{\lambda t}\right)\right)}^{\frac{x}{\lambda }}}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\right\},\\ =\underset{\lambda \to 0}{lim}\left(\frac{2\mathit{\lambda t}}{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right)\underset{\lambda \to 0}{lim}\left(\frac{{\left(1+log\left(1+\mathit{\lambda t}\right)\right)}^{\frac{x}{\lambda }}}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\right)\\ =\left(\frac{2t}{2+t}\right)\frac{{\left(1+t\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}\end{array}$$

In the same way, we can also define higher-order degenerate poly-Changhee polynomial by the following generating function:

$${\left(\frac{2\mathit{\lambda t}}{2\lambda +log\left(1+\mathit{\lambda t}\right)}\right)}^{\alpha }\frac{{\left(1+log{\left(1+\mathit{\lambda t}\right)}^{\frac{1}{\lambda }}\right)}^x}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}=\sum _{n=0}^{\infty }{\mathit{Ch}}_{n,\alpha ,\lambda }^{\left(\mathrm{k}\right)}\left(x\right)\frac{t^n}{n!},$$E31

where, $n\in {\mathbb{N}}_0,\mathrm{k}\in \mathbb{Z}$. After substituting $x=0$ in Eq. (30), we get higher-order degenerate poly-Changhee numbers, given as:

$${\left(\frac{2\mathit{\lambda t}}{2\lambda +log\left(1+t\right)}\right)}^{\alpha }\frac{1}{{\mathrm{Li}}_{\mathrm{k}}\left(1-e^{-t}\right)}=\sum _{n=0}^{\infty }{\mathit{Ch}}_{n,\alpha ,\lambda }^{\left(\mathrm{k}\right)}\left(0\right)\frac{t^n}{n!}.$$E32

Thus, one can see easily the more general results have been obtianed in this chapter.

Open Question: The authors suggest that the readers try to think about how to find generating function of various types of Apostol-based poly-Changhee polynomials, Bernoulli-based poly-Changhee polynomials, and Euler-based poly-Changhee polynomials?

Classifications

2010 Mathematics Subject Classification, 33C20, 33C45, 65A05.