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Some explicit identities on Changhee-Genocchi polynomials and numbers

Advances in Difference Equations20162016:202

https://doi.org/10.1186/s13662-016-0925-0

  • Received: 25 June 2016
  • Accepted: 25 July 2016
  • Published:

Abstract

In this paper, we introduce a new family of functions, which is called the Changhee-Genocchi polynomials. We study some explicit identities on these polynomials, which are related to Genocchi polynomials and Changhee polynomials. Also, we represent Changhee-Genocchi polynomials by gamma and beta functions.

We also study some properties of higher-order Changhee-Genocchi polynomials related to Changhee polynomials and Daehee polynomials.

Keywords

  • Euler polynomials
  • Changhee polynomials
  • Genocchi polynomials
  • Changhee-Genocchi numbers
  • beta and gamma functions

MSC

  • 05A10
  • 05A19
  • 11B68
  • 11S80

1 Introduction

The Genocchi polynomials are defined by the generating function (see [1, 2])
$$ \frac{2t}{e^{t}+1} e^{xt} = \sum _{n}^{\infty}G_{n}(x) \frac{t^{n}}{n!}. $$
(1)
When \(x=0\), \(G_{n}=G_{n}(0)\) are called the Genocchi numbers. From (1) we see that
$$\begin{aligned} \sum_{n=0}^{\infty}G_{n}(x) \frac{t^{n}}{n!} &= \biggl(\frac{2t}{e^{t}+1} \biggr) e^{xt} = \Biggl( \sum_{l=0}^{\infty}G_{l} \frac{t^{l}}{l!} \Biggr) \Biggl( \sum_{m=0}^{\infty}x^{m} \frac{t^{m}}{m!} \Biggr) \\ &= \sum_{n=0}^{\infty}\Biggl( \sum _{l=0}^{n} {n \choose l} G_{l} x^{n-l} \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(2)
We consider Changhee-Genocchi polynomials defined by the generating function
$$ \frac{2\log(1+t)}{2+t} (1+t)^{x} = \sum _{n=0}^{\infty}CG_{n}(x) \frac{t^{n}}{n!}. $$
(3)
When \(x=0\), \(CG_{n} = CG_{n}(0)\) are called the Changhee-Genocchi numbers.
The gamma and beta functions are defined by the following definite integrals:
$$ \Gamma(\alpha) = \int_{0}^{\infty}e^{-t} t^{\alpha-1}\,dt,\quad \alpha>0, $$
(4)
and
$$\begin{aligned} B(\alpha, \beta) &= \int_{0}^{1} t^{\alpha-1}(1-t)^{\beta-1}\,dt \\ &= \int_{0}^{\infty}\frac{t^{\alpha-1}}{(1+t)^{\alpha+\beta}}\,dt,\quad \alpha>0,\beta>0. \end{aligned}$$
(5)
From (4) and (5) we have (see [3])
$$ \Gamma(\alpha+1) = \alpha\Gamma(\alpha),\qquad B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha +\beta)}. $$
(6)
We recall that the classical Stirling numbers of the first kind \(S_{1}(n,k)\) and \(S_{2}(n,k)\) are defined by the relations (see [4])
$$\begin{aligned} &(x)_{n} = \sum_{k=0}^{n} S_{1}(n,k) x^{k} \quad\mbox{and}\\ &x^{n} = \sum_{k=0}^{n} S_{2}(n,k) (x)_{k}, \end{aligned}$$
respectively. Here \((x)_{n} = x(x-1)\cdots(x-n+1)\) denotes the falling factorial polynomial of order n. We also have
$$ \begin{aligned} &\sum_{n=m}^{\infty}S_{2}(n,m) \frac{t^{n}}{n!} = \frac{(e^{t}-1)^{m}}{m!} \quad\mbox{and}\\ &\sum_{n=m}^{\infty}S_{1}(n,m) \frac{t^{n}}{n!} = \frac{(\log(1+t))^{m}}{m!}. \end{aligned} $$
(7)

In this paper, we introduce a new family of functions, which is called the Changhee-Genocchi polynomials.

We study some properties of these polynomials, which are related to Genocchi polynomials and Changhee polynomials. Also we represent Changhee-Genocchi polynomials by gamma and beta functions.

We also study higher-order Changhee-Genocchi polynomials related to Changhee polynomials and Daehee polynomials.

Most of the ideas in this paper come from Kim and Kim [5]. Specifically, equations (14), (21), and (22) are related to the papers [58].

2 Changhee-Genocchi polynomials

First, we relate our newly defined Changhee-Genocchi polynomials to Genocchi polynomials.

Replacing t by \(e^{t}-1\) in (3) and applying (7), we get
$$\begin{aligned} \frac{2t}{e^{t}+1} e^{tx} &= \sum _{n=0}^{\infty}CG_{n}(x) \frac{1}{n!} \bigl(e^{t}-1\bigr)^{n} \\ &= \sum_{n=0}^{\infty}CG_{n}(x) \frac{1}{n!} n! \sum_{k=n}^{\infty}S_{2}(k,n) \frac{t^{k}}{k!} \\ &= \sum_{k=0}^{\infty}\Biggl( \sum _{n=0}^{k} CG_{n}(x) S_{2}(k,n) \Biggr)\frac{t^{k}}{k!}. \end{aligned}$$
(8)

The left-hand side of (8) is the generating function of the Genocchi polynomials.

Thus, by comparing the coefficients of (1) and (8) we have the following theorem.

Theorem 1

For any nonnegative integer k, we have
$$ G_{k}(x) = \sum_{n=0}^{k} CG_{n}(x) S_{2}(k,n). $$
(9)
On the other hand, if we replace t by \(\log(1+t)\) in (1) and apply (7), then we get
$$\begin{aligned} \frac{2\log(1+t)}{2+t} (1+t)^{x} &= \sum_{n=0}^{\infty}G_{n}(x) \frac{1}{n!} \bigl( \log(1+t) \bigr)^{n} \\ &= \sum_{n=0}^{\infty}G_{n}(x) \frac{1}{n!} n! \sum_{k=n}^{\infty}S_{1}(k,n) \frac{t^{k}}{k!} \\ &= \sum_{k=0}^{\infty}\Biggl( \sum _{n=0}^{k} G_{n}(x) S_{1}(k,n) \Biggr) \frac{t^{k}}{k!}, \end{aligned}$$
(10)
where \(S_{1}(k,n)\) are the Stirling numbers of the first kind.

By comparing the coefficients of both sides of (10), we get the following theorem.

Theorem 2

For any nonnegative integer k, we have
$$ CG_{k}(x) = \sum_{n=0}^{k} G_{n}(x) S_{1}(k,n). $$
(11)

Remark

When \(x=0\) in (11), we can see that Changhee-Genocchi numbers are integers.

We can consider equation (11) as the inversion formula for (9). From (3) we can consider the following identity:
$$\begin{aligned} \sum_{n=0}^{\infty}CG_{n}(x) \frac{t^{n}}{n!} &= \frac{2\log(1+t)}{2+t} (1+t)^{x} = \Biggl( \sum_{l=0}^{\infty}CG_{l} \frac{t^{l}}{l!} \Biggr) \Biggl(\sum_{m=0}^{\infty}(x)_{m} \frac{t^{m}}{m!} \Biggr) \\ &= \sum_{n=0}^{\infty}\Biggl(\sum _{l=0}^{n}{n \choose l}CG_{l}(x)_{n-l} \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(12)
Thus, by comparing the coefficients of both sides of (12) we have
$$\begin{aligned} CG_{n}(x) &= \sum _{l=0}^{n} {n \choose l} CG_{l} (x)_{n-l} = \sum_{l=0}^{n} {n \choose l} CG_{n-l} (x)_{l} \\ &= \sum_{l=0}^{n} \Biggl( \sum _{m=0}^{n-l}{n \choose l} CG_{l} S_{1}(n-l, m) x^{m} \Biggr). \end{aligned}$$
(13)
From (13) we can derive the following theorem.

Theorem 3

For any nonnegative integer n, we have
$$ \int_{0}^{1} CG_{n}(x)\,dx = \sum _{l=0}^{n}\sum_{m=0}^{n-l}{n \choose l} CG_{l} S_{1}(n-l, m) \frac{1}{m+1}. $$
(14)
In this paper, we define the λ-Changhee-Genocchi polynomials by a generating function as follows:
$$ \frac{2\log(1+t)}{(1+t)^{\lambda}+ 1} (1+t)^{\lambda x} = \sum _{n=0}^{\infty}CG_{n,\lambda} (x) \frac{t^{n}}{n!}. $$
(15)
We recall that the λ-Changhee polynomials are defined in [9] by
$$ \frac{2}{(1+t)^{\lambda}+ 1} (1+t)^{\lambda x} = \sum _{n=0}^{\infty}Ch_{n,\lambda}(x) \frac{t^{n}}{n!}. $$
(16)
When \(\lambda=1\), Changhee-Genocchi polynomials are well-known Changhee polynomials, cf. [1018]. In order to establish a reflexive symmetry on the Changhee-Genocchi polynomials, we consider the following:
$$\begin{aligned} \sum_{n=0}^{\infty}CG_{n}(1-x)\frac{t^{n}}{n!} &= \frac{2\log(1+t)}{1+(1+t)}(1+t)^{1-x} = -\frac{2\log(1+t)}{(1+t)^{-1}+1}(1+t)^{-x} \\ &= \sum_{n=0}^{\infty}CG_{n,-1}(x) \frac{t^{n}}{n!}. \end{aligned}$$
(17)
By comparing the coefficients of (17) we have the following theorem.

Theorem 4

For \(n\in\mathbb {N}\), we have
$$ CG_{n}(1-x) = CG_{n,-1}(x). $$
(18)
Thus, from (3) and (18) we have
$$\begin{aligned} \sum_{n=0}^{\infty}CG_{n}\bigl(-x+(1-y)\bigr)\frac{t^{n}}{n!} &= \frac{2\log(1+t)}{2+t}(1+t)^{-x+(1-y)} \\ &= \frac{2\log(1+t)}{2+t}(1+t)^{-x}(1+t)^{1-y} \\ &= \Biggl(\sum_{m=0}^{\infty}CG_{m}(-x)\frac{t^{m}}{m!} \Biggr) \Biggl(\sum _{l=0}^{\infty}(1-y)_{l}(-x)\frac{t^{l}}{l!} \Biggr) \\ &= \sum_{n=0}^{\infty}\Biggl( \sum _{m=0}^{n}{n\choose m} CG_{m}(-x) (1-y)_{n-m} \Biggr)\frac{t^{n}}{n!} \\ &= \sum_{n=0}^{\infty}\sum _{m=0}^{n} \sum_{k=0}^{n-m} {n\choose m} CG_{m}(-x)S_{1}(n-m, k) (1-y)^{k}. \end{aligned}$$
(19)
By comparing the coefficients of (19) we have
$$ CG_{n}\bigl(1-(x+y)\bigr) = \sum _{m=0}^{n}\sum_{k=0}^{n-m}{n \choose m}CG_{m}(-x) S_{1}(n-m, k) (1-y)^{k}. $$
(20)
On the other hand, by (5), (6), and (20) we have
$$\begin{aligned} &\int_{0}^{1} y^{n} CG_{n} \bigl(1-(x+y)\bigr)\,dy \\ &\quad= \sum_{m=0}^{n}\sum _{k=0}^{n-m}{n\choose m}CG_{m}(-x) S_{1}(n-m, k) B(n+1, k+1) \\ &\quad= \sum_{m=0}^{n}\sum _{k=0}^{n-m}{n\choose m}CG_{m}(-x)S_{1}(n-m,k) \frac{\Gamma(n+1)\Gamma(k+1)}{\Gamma(n+k+2)}. \end{aligned}$$
(21)
Thus, by (18) and (21) we have the following identities, which relate the λ-Changhee-Genocchi polynomials, the Stirling numbers, and the beta and gamma polynomials:
$$\begin{aligned} &\int_{0}^{1} y^{n} CG_{n,-1}(x+y)\,dy \\ &\quad= -\sum_{l=0}^{n}\sum _{m=0}^{n-l}{n\choose l}S_{1}(n-l,m)CG_{l} \int_{0}^{1} y^{n} \bigl(1-(x+y) \bigr)^{m} \,dy \\ &\quad= -\sum_{l=0}^{n}\sum _{m=0}^{n-l}\sum_{k=0}^{m}{n \choose l} {m\choose k}S_{1}(n-l,m) (-x)^{m-k} CG_{l} \int_{0}^{1} y^{n} (1-y)^{k} \,dy \\ &\quad= -\sum_{l=0}^{n}\sum _{m=0}^{n-l}\sum_{k=0}^{m}{n \choose l} {m\choose k}S_{1}(n-l,m) (-x)^{m-k} CG_{l} B(n+1, k+1) \\ &\quad= -\sum_{l=0}^{n}\sum _{m=0}^{n-l}\sum_{k=0}^{m}{n \choose l} {m\choose k}S_{1}(n-l,m) (-x)^{m-k} CG_{l} \frac{\Gamma(n+1)\Gamma(k+1)}{\Gamma(n+k+2)}. \end{aligned}$$
(22)
From (16) we consider
$$\begin{aligned} \sum_{n=0}^{\infty}CG_{n,\lambda}(1-x) \frac{t^{n}}{n!} &= \frac{2\log(1+t)}{(1+t)^{\lambda}+ 1}(1+t)^{\lambda(1-x)} = \frac{2\log(1+t)}{1+(1+t)^{-\lambda}}(1+t)^{-\lambda x} \\ &= \sum_{n=0}^{\infty}CG_{n,-\lambda}(x) \frac{t^{n}}{n!}. \end{aligned}$$
(23)
By comparing the coefficients of (23) we have the following theorem.

Theorem 5

For any nonnegative integer n, we have
$$ CG_{n,\lambda}(1-x) = CG_{n,-\lambda}(x). $$
(24)

Remark

If we take \(\lambda=1\) in Theorem 5, then we have the result in Theorem 4.

From the second line of (23) and from (16) we have
$$\begin{aligned} & \Biggl( \sum _{l=1}^{\infty}\frac{(-1)^{l-1} t^{l}}{l} \Biggr) \Biggl( \sum _{m=0}^{\infty}Ch_{m,\lambda}(x) \frac{t^{m}}{m!} \Biggr) \\ &\quad= \sum_{n=1}^{\infty} \Biggl( \sum _{l=1}^{n} \frac{(-1)^{l-1}}{l} \frac{Ch_{n-l,\lambda}(x)}{(n-l)!}n! \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(25)

By comparing the coefficients of (23) and (25) we have the following theorem.

Theorem 6

For any positive integer n, we have
$$ CG_{n,\lambda}(x) = \sum_{l=1}^{n} \frac{(-1)^{l-1}}{l} Ch_{n-l,\lambda }(x)\frac{n!}{(n-l)!}. $$
For \(r\in\mathbb {N}\), we define the Changhee-Genocchi polynomials \(CG_{n}^{(r)}(x)\) of order r by the generating function
$$ \biggl( \frac{2\log(1+t)}{2+t} \biggr)^{r} (1+t)^{x} = \sum_{n=0}^{\infty}CG_{n}^{(r)}(x)\frac{t^{n}}{n!}. $$
(26)
From (26) we have the following relation between the Changhee-Genocchi polynomials of order r and the Changhee polynomials of order r:
$$\begin{aligned} &\bigl(\log(1+t) \bigr)^{r} \biggl(\frac{2}{2+t} \biggr)^{r} (1+t)^{x} \\ &\quad= \Biggl( r!\sum_{l=r}^{\infty}S_{2}(l,r)\frac{t^{l}}{l!} \Biggr) \Biggl( \sum _{m=0}^{\infty}Ch_{m}^{(r)}(x) \frac{t^{m}}{m!} \Biggr) \\ &\quad= \Biggl( \sum_{l=0}^{\infty}S_{2}(l+r,r)\frac{r! t^{l+r}}{(l+r)!} \Biggr) \Biggl( \sum _{m=0}^{\infty}Ch_{m}^{(r)}(x) \frac{t^{m}}{m!} \Biggr) \\ &\quad= \Biggl( \sum_{l=0}^{\infty}S_{2}(l+r,r) {l+r \choose r}^{-1} \frac {t^{l}}{l!} \Biggr) \Biggl( \sum_{m=0}^{\infty}Ch_{m}^{(r)}(x) \frac{t^{m}}{m!} \Biggr) t^{r} \\ &\quad= \sum_{n=0}^{\infty}\Biggl( \sum _{l=0}^{n} {n\choose l} S_{2}(l+r,r){l+r \choose r}^{-1} Ch_{n-l}^{(r)}(x) \Biggr) \frac{t^{n+r}}{n!}. \end{aligned}$$
(27)
By comparing the coefficients of (26) and (27) we have the following theorem.

Theorem 7

For any nonnegative integer n, we have
$$ CG_{n}^{(r)}(x) = \sum_{l=0}^{n}{n \choose l} {l+r\choose r}^{-1}S_{2}(l+r,r)Ch_{n-l}^{(r)}(x). $$
For \(d\in\mathbb {N}\) with \(d\equiv1\ (\operatorname{mod}2)\), we have the following identity:
$$ \sum_{a=0}^{d-1}(-1)^{a}(1+t)^{a} = \frac{1+(1+t)^{d}}{2+t}. $$
(28)
So, for such \(d\equiv1\ (\operatorname{mod} 2)\), from (28), (3), and (15) we see that
$$\begin{aligned} \sum_{n=0}^{\infty}CG_{n}(x)\frac{t^{n}}{n!} &= \frac{2\log (1+t)}{2+t}(1+t)^{x} \\ &= \sum_{a=0}^{d-1}(-1)^{a} \frac{2\log(1+t)}{(1+t)^{d}+1}(1+t)^{d (\frac{a+x}{d} )} \\ &= \sum_{a=0}^{d-1}(-1)^{a}\sum _{n=0}^{\infty}CG_{n,d} \biggl( \frac {a+x}{d} \biggr)\frac{t^{n}}{n!} \\ &= \sum_{n=0}^{\infty}\Biggl(\sum _{a=0}^{d-1}(-1)^{a} CG_{n,d} \biggl(\frac {a+x}{d} \biggr) \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(29)
By comparing the coefficients in (29), for \(d\equiv1\ (\operatorname{mod} 2)\), we have the following theorem.

Theorem 8

For any nonnegative integer n and \(d\equiv1\ (\operatorname{mod} 2)\), we have
$$ CG_{n}(x) = \sum_{a=0}^{d-1} (-1)^{a} CG_{n,d} \biggl(\frac{a+x}{d} \biggr). $$
(30)

We remark that, for \(d\equiv1\ (\operatorname{mod} 2)\), from (9) and (30) we have the inversion of Theorem 8.

Theorem 9

For any nonnegative integer n and \(d\equiv1\ (\operatorname{mod} 2)\), we have
$$\begin{aligned} G_{k}(x) &= \sum_{n=0}^{k} CG_{n}(x) S_{2}(k,n) \\ &= \sum_{n=0}^{k} \Biggl( \sum _{a=0}^{d-1} (-1)^{a} CG_{n,d} \biggl(\frac {a+x}{d} \biggr) S_{2}(k,n) \Biggr). \end{aligned}$$
From the generating function of the Changhee-Genocchi polynomials in (1), replacing t by \(\lambda\log(1+t)\), we get
$$\begin{aligned} \frac{2\lambda\log(1+t)}{(1+t)^{\lambda}+1}(1+t)^{\lambda x} &= \sum_{n=0}^{\infty}G_{n}(x) \frac{1}{n!} \bigl( \lambda\log(1+t) \bigr)^{n} \\ &= \sum_{n=0}^{\infty}\lambda^{n} G_{n}(x) \Biggl( \sum_{k=n}^{\infty}S_{1}(k,n)\frac{t^{k}}{k!} \Biggr) \\ &= \sum_{k=0}^{\infty}\Biggl( \sum _{n=0}^{k}\lambda^{n} G_{n}(x) S_{1}(k,n) \Biggr)\frac{t^{k}}{k!}. \end{aligned}$$
(31)
Thus, the left-hand side of (31) can be represented by the λ-Changhee-Genocchi polynomials as follows:
$$ \frac{2\lambda\log(1+t)}{(1+t)^{\lambda}+1} (1+t)^{\lambda x} = \lambda\sum _{k=0}^{\infty}CG_{k,\lambda}(x)\frac{t^{k}}{k!}. $$
(32)
By comparing the coefficients of (31) and (32) we have the following theorem.

Theorem 10

For any nonnegative integer k, we have
$$ CG_{k,\lambda}(x) = \sum_{n=0}^{k} \lambda^{n-1} G_{n}(x) S_{1}(k,n). $$
From the generating function of the Changhee-Genocchi numbers in (3) we want to see the recurrence relation for the Changhee-Genocchi numbers:
$$\begin{aligned} 2\log(1+t) &= \sum _{n=0}^{\infty}CG_{n} \frac{t^{n}}{n!}(t+2) \\ &= \sum_{n=1}^{\infty}CG_{n} \frac{t^{n+1}}{n!} + \sum_{n=0}^{\infty}2 CG_{n} \frac{t^{n}}{n!} \\ &= \sum_{n=2}^{\infty}n CG_{n-1} \frac{t^{n}}{n!} + 2\sum_{n=1}^{\infty}CG_{n} \frac{t^{n}}{n!} \\ &= 2CG_{1} t + \sum_{n=2}^{\infty}(n CG_{n-1} + 2CG_{n})\frac{t^{n}}{n!}. \end{aligned}$$
(33)
On the other hand, from the left-hand side of (33) we have
$$ 2\log(1+t) = \sum_{n=1}^{\infty}(-1)^{n-1} 2(n-1)! \frac{t^{n}}{n!}. $$
(34)
By comparing the coefficients of (33) and (34) we have the following recurrence relation for the Changhee-Genocchi numbers.

Theorem 11

We have
$$\begin{aligned} & CG_{0} = 0,\\ & nCG_{n-1} + 2CG_{n} = 2(n-1)!(-1)^{n-1} \quad\textit{for } n\ge1. \end{aligned}$$
From the higher-order Changhee-Genocchi polynomials
$$ \biggl( \frac{2\log(1+t)}{2+t} \biggr)^{r}(1+t)^{x} = \sum_{n=0}^{\infty}CG_{n}^{(r)}(x) \frac{t^{n}}{n!} $$
(35)
we can deduce
$$ CG_{0}^{(r)}(x) = CG_{1}^{(r)}(x) = \cdots= CG_{r-1}^{(r)}(x) = 0. $$
(36)
Thus, from (36) we can rewrite (35) as follows:
$$ \biggl( \frac{2\log(1+t)}{2+t} \biggr)^{r}(1+t)^{x} = \sum_{n=0}^{\infty}CG_{n+r}^{(r)}(x) \frac{t^{n+r}}{(n+r)!}. $$
(37)
We recall that the Dahee polynomials are defined by the generating function (see [9, 19])
$$ \frac{\log(1+t)}{t} (1+t)^{x} = \sum_{n=0}^{\infty}D_{n}(x) \frac{t^{n}}{n!}. $$
When \(x=0\), \(D_{n} = D_{n}(0)\) are called the Dahee numbers.
For \(r\in\mathbb {N}\), the higher-order Daehee numbers are given by the generating function (see [9, 19, 20])
$$ \biggl(\frac{\log(1+t)}{t} \biggr)^{r} = \sum _{n=0}^{\infty}D_{n}^{(r)}(x) \frac{t^{n}}{n!}. $$
From (28) we have
$$\begin{aligned} 2\log(1+t)\sum _{a=0}^{d-1}(-1)^{a}(1+t)^{a} &= \frac{2\log(1+t)}{2+t} + \frac{2\log(1+t)}{t+2}(1+t)^{d} \\ &= \frac{2\log(1+t)}{t} \Biggl( \sum_{a=0}^{d-1}(-1)^{a}(1+t)^{a} \Biggr) \\ &= \sum_{n=0}^{\infty}CG_{n} \frac{t^{n-1}}{n!} + \sum_{n=0}^{\infty}CG_{n}(d)\frac{t^{n-1}}{n!} \\ &= \sum_{n=0}^{\infty}\Biggl( 2\sum _{a=0}^{d-1}(-1)^{a} D_{n}(a) \Biggr) \frac{t^{n}}{n!} \\ &= \sum_{n=0}^{\infty}\biggl( \frac{CG_{n+1}}{n+1} + \frac {CG_{n+1}(d)}{n+1} \biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(38)
Thus, from (38) we have the following theorem.

Theorem 12

For any nonnegative integer n and \(d\equiv1\ (\operatorname{mod} 2)\), we have
$$ 2\sum_{a=0}^{d-1}(-1)^{a} D_{n}(a) = \frac{CG_{n+1}}{n+1} + \frac{CG_{n+1,d}}{n+1}. $$

3 Changhee-Genocchi polynomials arising from differential equations

In this section, we give new identities on the Changhee-Genocchi numbers by using differential equations. We use the idea recently developed by Kwon et al. [21].

By equation (3) we can write the generating function for the Changhee-Genocchi numbers as follows:
$$ F(t) = \frac{2\log(1+t)}{2+t} = \sum_{n=0}^{\infty}CG_{n}\frac{t^{n}}{n!}. $$
(39)
Let
$$\begin{aligned} G(t) = \log(1+t) \quad\mbox{and}\quad H(t) = \frac{2}{2+t}. \end{aligned}$$
Then
$$\begin{aligned} G^{(N)}(t) &= \biggl(\frac{d}{dt} \biggr)^{N} G(t) = (-1)^{N-1}(N-1)! e^{-N\cdot G(t)}, \quad\mbox{and}\\ H^{(N)}(t) &= \biggl(\frac{d}{dt} \biggr)^{N} H(t)\\ &= \biggl(-\frac{1}{2} \biggr)^{N} N! e^{-(N+1)\cdot K(t)},\quad \mbox{where } K(t)= \log(1+t/2). \end{aligned}$$
Thus,
$$\begin{aligned} F^{(N)}(t) ={}& \biggl( \frac{d}{dt} \biggr)^{N} F(t) = \sum _{k=0}^{N}{N\choose k}G^{(N-k)}H^{(k)} \\ ={}& \sum_{k=0}^{N} {N\choose k} (-1)^{N-k-1} (N-k-1)! e^{-(N-k)G(t)} \\ &{} \times \biggl(-\frac{1}{2} \biggr)^{k} k! e^{-(k+1)K(t)} \\ ={}& \sum_{k=0}^{N} {N\choose k} (-1)^{N-1} \biggl(\frac{1}{2} \biggr)^{k} k! (N-k-1)! e^{-(N-k)G(t)} e^{-(k+1)K(t)}. \end{aligned}$$
(40)
On the other hand,
$$\begin{aligned} e^{-(N-k)G} e^{-(k+1)K} ={}& \Biggl( \sum_{n=0}^{\infty}(-N+k)^{n} \frac {G^{n}}{n!} \Biggr) \Biggl( \sum_{l=0}^{\infty}\bigl(-(k+1)\bigr)^{l}\frac{K^{l}}{l!} \Biggr) \\ ={}& \Biggl( \sum_{n=0}^{\infty}(-N+k)^{n} \sum_{m=n}^{\infty}S_{1}(m,n) \frac {t^{m}}{m!} \Biggr) \\ &{} \times \Biggl( \sum_{l=0}^{\infty}(-k-1)^{l} \sum_{j=l}^{\infty}\frac{1}{2^{j}} S_{1}(j,l)\frac{t^{j}}{j!} \Biggr) \\ ={}& \sum_{m=0}^{\infty}\Biggl(\sum _{n=0}^{m}(-N+k)^{n} S_{1}(m,n) \Biggr)\frac {t^{m}}{m!} \\ &{} \times\sum_{j=0}^{\infty}\Biggl(\sum_{l=0}^{j}(-k-1)^{l} S_{1}(j,l)\frac{1}{2^{j}} \Biggr)\frac{t^{j}}{j!} \\ ={}& \sum_{s=0}^{\infty}\Biggl( \sum _{m=0}^{s}{s\choose m} \sum _{n=0}^{m}(-N+k)^{n} S_{1}(m,n) \\ &{}\times\sum_{l=0}^{s-m}(-k-1)^{l} S_{1}(s-m,l) \frac {1}{2^{s-m}} \Biggr)\frac{t^{s}}{s!}. \end{aligned}$$
(41)
From (39) we have
$$ F^{(N)}(t) = \biggl(\frac{d}{dt} \biggr)^{N} F(t) = \sum_{m=0}^{\infty}CG_{N+m} \frac{t^{m}}{m!}. $$
(42)
By comparing the coefficients of (40), (41), and (42) we have new identities on the Changhee-Genocchi numbers as follows.

Theorem 13

For any nonnegative integer s, we have
$$\begin{aligned} CG_{s+N} ={}& \sum_{m=0}^{s}{s \choose m} \Biggl\{ \Biggl( \sum_{n=0}^{m}(-N+k)^{n} S_{1}(m,n) \Biggr) \Biggl( \sum_{l=0}^{s-m}(-k-1)^{l} S_{1}(s-m, l)\frac{1}{2^{s-m}} \Biggr) \Biggr\} \\ &{} \times\sum_{k=0}^{N} {N \choose k} (-1)^{N-1} \biggl(\frac {1}{2} \biggr)^{k} k! (N-k-1)!. \end{aligned}$$

Declarations

Acknowledgements

The authors would like to express their sincere gratitude to the Editor, who gave us valuable comments to improve this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mechanical System Engineering, Dongguk University, 123 Dongdae-ro, Gyungju-si, Gyeongsangbuk-do, 38066, S. Korea
(2)
Department of Mathematics Education, Kyungpook National University, 80 Daehakro, Bukgu, Daegu, 41566, S. Korea

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