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Ordinary and degenerate Euler numbers and polynomials

  • Taekyun Kim
  • Dae San Kim
  • Han Young Kim
  • Jongkyum KwonEmail author
Open Access
Research
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Abstract

In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on \(\mathbb{Z}_{p}\). Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

Keywords

Euler polynomials and numbers Degenerate Euler polynomials and numbers Alternating integer power sum polynomials Degenerate alternating integer power sum polynomials 

MSC

11B68 11B83 11S80 

1 Introduction

Let p be a fixed odd prime number. Throughout this paper, \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\), and \(\mathbb{C}_{p}\) will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of \(\mathbb{Q}_{p}\), respectively. The p-adic norm is normalized as \(|p|_{p}=\frac{1}{p}\).

Let f be a \(\mathbb{C}_{p}\)-valued continuous function on \(\mathbb{Z}_{p}\). Then the fermionic p-adic integral of f on \(\mathbb{Z}_{p}\) is defined by Kim as
$$ \begin{aligned}[b] \int _{\mathbb{Z}_{p}} f(x)\,d\mu _{-1}(x) & = \lim _{N\rightarrow \infty } \sum_{x=0}^{p^{N}-1}f(x) \mu _{-1}\bigl(x+p^{N} \mathbb{Z}_{p} \bigr) \\ & = \lim_{N\rightarrow \infty } \sum_{x=0}^{p^{N}-1}f(x) (-1)^{x} \quad (\text{see [11, 12]}). \end{aligned} $$
(1.1)
From (1.1), we note that
$$ \int _{\mathbb{Z}_{p}} f(x+1)\,d\mu _{-1}(x) = - \int _{\mathbb{Z}_{p}} f(x)\,d \mu _{-1}(x) + 2 f(0)\quad ( \text{see [5, 8]}), $$
(1.2)
and by induction, for any \(n \in \mathbb{N}\), we get
$$ \int _{\mathbb{Z}_{p}} f(x+n)\,du_{-1}(x) = (-1)^{n} \int _{\mathbb{Z} _{p}} f(x)\,du_{-1}(x) + 2 \sum _{l=0}^{n-1}(-1)^{n-1-l} f(l). $$
(1.3)
It is well known that the Euler polynomials are defined by
$$ \frac{2}{e^{t}+1} e^{xt} = \sum _{n=0}^{\infty }E_{n}(x) \frac{t^{n}}{n!}\quad ( \text{see [1--21]}). $$
(1.4)
When \(x=0\), \(E_{n} = E_{n}(0)\) are called the Euler numbers.
From (1.4), we note that
$$ E_{n}(x) = \sum_{l=0}^{n} \binom{n}{l} E_{l} x^{n-l},\quad n \geq 0 \mbox{ } (\text{see [1--21]}), $$
(1.5)
where n is a nonnegative integer.
By (1.4) and (1.5), we get
$$ \begin{aligned}[b] E_{n}(1)+E_{n}= \sum _{l=0}^{n} \binom{n}{l} E_{l} + E_{n} = \textstyle\begin{cases} 2, & \text{if } n = 0, \\ 0, & \text{if } n > 0. \end{cases}\displaystyle \end{aligned} $$
(1.6)
Let
$$ T_{p} (n) = 2 \sum_{k=1}^{n} (-1)^{k-1} k^{p},\quad n, p \in \mathbb{N}. $$
(1.7)
Then, by (1.4) and (1.5), we get
$$ \begin{aligned}[b] \sum_{p=0}^{\infty }T_{p}(n) \frac{t^{p}}{p!} & = 2 \sum_{k=1}^{n} (-1)^{k-1} e^{kt} = \frac{2}{e^{t}+1} \bigl(e^{(n+1)t} + e^{t} \bigr) \\ & = \sum_{p=0}^{\infty } \bigl(E_{p}(n+1) + E_{p}(1) \bigr) \frac{t ^{p}}{p!}, \end{aligned} $$
(1.8)
where \(n \in \mathbb{N}\), with \(n \equiv 1~(\mathrm{mod}~ 2)\). Thus we have, for \(n,p \in \mathbb{N}\), with \(n \equiv 1~(\mathrm{mod}~2)\),
$$ T_{p} (n) = E_{p}(n+1) - E_{p}. $$
(1.9)
From (1.2), we can derive the following equation (1.10):
$$ \int _{\mathbb{Z}_{p}} e^{(x+y)t} \,d\mu _{-1}(y) = \frac{2}{e^{t}+1} e ^{xt} = \sum_{n=0}^{\infty }E_{n}(x) \frac{t^{n}}{n!}. $$
(1.10)
Thus, by (1.10), we get
$$ \int _{\mathbb{Z}_{p}} (x+y)^{n} \,d\mu _{-1}(y) = E_{n}(x),\quad n \geq 0\mbox{ } (\text{see [11]}). $$
(1.11)
Thus, by (1.9) and (1.11), we have
$$ T_{p}(n) = \int _{\mathbb{Z}_{p}} (x+n+1)^{p} \,d\mu _{-1}(x) - \int _{\mathbb{Z}_{p}} x^{p} \,d\mu _{-1}(x), $$
(1.12)
where \(n,p \in \mathbb{N}\) with \(n \equiv 1~(\mathrm{mod}~ 2)\).
We recall here that the Stirling numbers of the second kind are given by the exponential generating function
$$ \frac{1}{k!}\bigl(e^{t} -1\bigr)^{k}= \sum_{n=k}^{\infty }S_{2}(n,k) \frac{t^{n}}{n!}. $$
(1.13)

The purpose of this paper is to investigate some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on \(\mathbb{Z}_{p}\).

The outline of this paper is as follows. In Sect. 1, we will review some necessary results about fermionic p-adic integrals, Euler polynomials, and alternating integer power sums. In Sect. 2, we will introduce the alternating integer power sum polynomials and represent them in terms of Euler polynomials and Stirling numbers of the second kind, and derive various properties about Euler numbers and polynomials. In Sect. 3, we will introduce the degenerate alternating integer power sum polynomials and express them in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second, and derive some properties on degenerate Euler numbers and polynomials.

2 Some identities of Euler numbers and polynomials

In this section, we will introduce the alternating integer power sum polynomials and represent them in terms of Euler polynomials and Stirling numbers of the second kind, and derive various properties about Euler numbers and polynomials.

For \(p \in \mathbb{N}\), we have
$$ \begin{aligned}[b] (-1)^{j} (j+1)^{p} +(-1)^{j} j^{p} & = \sum _{i=0}^{p} \binom{p}{i} j ^{i} (-1)^{j} + (-1)^{j} j^{p} \\ & = 2 (-1)^{j} j^{p} + \sum_{i=1}^{p-1} \binom{p}{i} j^{i} (-1)^{j} + (-1)^{j}. \end{aligned} $$
(2.1)
From (2.1), for \(n,p \in \mathbb{N}\) with \(n \equiv 1~( \mathrm{mod}~ 2)\), we note that
$$ \begin{aligned}[b] (-1)^{n} (n+1)^{p} & = \sum_{j=0}^{n} \bigl\{ (-1)^{j} (j+1)^{p} + (-1)^{j} j^{p} \bigr\} \\ & = 2 \sum_{j=0}^{n} (-1)^{j} j^{p} + \sum_{i=1}^{p-1} \binom{p}{i} \sum_{j=0}^{n} (-1)^{j} j^{i} + \sum_{j=0}^{n} (-1)^{j} \\ & = 2 \sum_{j=1}^{n} (-1)^{j} j^{p} + \sum_{i=1}^{p-1} \binom{p}{i} \sum_{j=1}^{n} (-1)^{j} j^{i}. \end{aligned} $$
(2.2)
By (1.7) and (2.2), we get
$$ T_{p}(n) = (n+1)^{p} -\frac{1}{2} \sum_{i=1}^{p-1} \binom{p}{i} T_{i}(n), $$
(2.3)
where \(n,p \in \mathbb{N}\) with \(n \equiv 1~(\mathrm{mod}~ 2)\).

Therefore, by (2.3), we obtain the following theorem.

Theorem 2.1

Let\(n,p \in \mathbb{N}\)with\(n \equiv 1~(\mathrm{mod}~ 2)\). Then we have
$$ \begin{aligned}[b] & \int _{\mathbb{Z}_{p}} (x+n+1)^{p} \,d\mu _{-1}(x) - \int _{\mathbb{Z} _{p}} x^{p} \,d\mu _{-1}(x) \\ &\quad = (n+1)^{p} - \frac{1}{2}\sum_{i=1}^{p-1} \binom{p}{i} \biggl\{ \int _{\mathbb{Z}_{p}} (x+n+1)^{i} \,d\mu _{-1}(x) - \int _{\mathbb{Z}_{p}} x^{i} \,d\mu _{-1}(x) \biggr\} . \end{aligned} $$
(2.4)

From (1.11) and Theorem 2.1, we note the following corollary.

Corollary 2.2

Let\(n,p \in \mathbb{N}\)with\(n \equiv 1~(\mathrm{mod}~ 2)\). Then we have
$$ E_{p}(n+1) - E_{p} = (n+1)^{p} - \frac{1}{2}\sum_{i=1}^{p-1} \binom{p}{i} \bigl(E_{i}(n+1) - E_{i} \bigr). $$
(2.5)
For \(n \in \mathbb{N}_{0} = \mathbb{N} \cup \{0\}\), we have
$$ \int _{\mathbb{Z}_{p}} (y+1-x)^{n} \,d\mu _{-1}(y) = (-1)^{n} \int _{\mathbb{Z}_{p}} (y+x)^{n} \,d\mu _{-1}(y). $$
(2.6)
Thus, by (2.6), we get
$$ E_{n}(1-x) = (-1)^{n} E_{n} (x),\quad n \geq 0. $$
For \(n \in \mathbb{N}_{0}\), and by (1.2), we have
$$ \begin{aligned}[b] \int _{\mathbb{Z}_{p}} (x+2)^{n} \,d\mu _{-1}(x) & = \sum _{l=0}^{n} \binom{n}{l} \int _{\mathbb{Z}_{p}} (x+1)^{l} \,d\mu _{-1}(x) \\ & = 1 + \sum_{l=1}^{n} \binom{n}{l} \int _{\mathbb{Z}_{p}} (x+1)^{l} \,d \mu _{-1}(x) \\ & = 1 - \sum_{l=1}^{n} \binom{n}{l} \int _{\mathbb{Z}_{p}} x^{l} \,d\mu _{-1}(x) \\ & = 2 - \sum_{l=0}^{n} \binom{n}{l} \int _{\mathbb{Z}_{p}} x^{l} \,d\mu _{-1}(x). \end{aligned} $$
(2.7)

Thus, by using (1.2) and (2.7), we get the next theorem.

Theorem 2.3

For\(n \in \mathbb{N} \cup \{0 \}\), we have
$$ \int _{\mathbb{Z}_{p}} (x+2)^{n} \,d\mu _{-1}(x) = 2 + \int _{\mathbb{Z} _{p}} x^{n} \,d\mu _{-1}(x) - 2 \delta _{0,n}, $$
where\(\delta _{n,k}\)is the Kronecker’s delta.

By combining Theorem 2.3 with (1.11), we arrive at the following corollary.

Corollary 2.4

For\(n \in \mathbb{N}\), we have
$$ E_{n}(2) = 2 + E_{n}. $$
(2.8)
For the next result, we note that, for any \(n \in \mathbb{N}\),
$$ E_{n} = (-1)^{n-1}E_{n}. $$
(2.9)
For \(m,n \in \mathbb{N}\), and by (1.11) and (2.9), we have
$$ \begin{aligned}[b] \int _{\mathbb{Z}_{p}} x^{m} (x-1)^{n} \,d\mu _{-1}(x) & = \sum_{i=0}^{n} \binom{n}{i} (-1)^{n-i} \int _{\mathbb{Z}_{p}} x^{m+i} \,d\mu _{-1}(x) \\ & = \sum_{i=0}^{n} \binom{n}{i} (-1)^{n-i} E_{m+i} \\ & = (-1)^{m+n-1} \sum_{i=0}^{n} \binom{n}{i} E_{m+i}. \end{aligned} $$
(2.10)
On the other hand, by (2.6) and (2.8), we get
$$ \begin{aligned}[b] \int _{\mathbb{Z}_{p}} x^{m} (x-1)^{n} \,d\mu _{-1}(x) & = \sum_{i=0}^{m} \binom{m}{i} \int _{\mathbb{Z}_{p}} (x-1)^{n+i} \,d\mu _{-1}(x) \\ & = \sum_{i=0}^{m} \binom{m}{i} (-1)^{n+i} \int _{\mathbb{Z}_{p}} (x+2)^{n+i} \,d\mu _{-1}(x) \\ & = \sum_{i=0}^{m} \binom{m}{i} (-1)^{n+i} (E_{n+i} +2 ) \\ & = \sum_{i=0}^{m} \binom{m}{i} (-1)^{n+i} E_{n+i} \\ & = - \sum_{i=0}^{m} \binom{m}{i} E_{n+i}. \end{aligned} $$
(2.11)

Therefore, by (2.10) and (2.11), we obtain the following theorem.

Theorem 2.5

For\(m,n \in \mathbb{N}\), the following symmetric identity holds:
$$ (-1)^{n} \sum_{i=0}^{n} \binom{n}{i} E_{m+i} = (-1)^{m} \sum _{i=0} ^{m} \binom{m}{i} E_{n+i}. $$
(2.12)
Now, we define the alternating integer power sum polynomials by
$$ T_{p}(n | x) = 2 \sum_{k=0}^{n} (-1)^{k-1} (k+x)^{p},\quad n , p \in \mathbb{N}_{0}. $$
(2.13)

Note that \(T_{p}(n | 0) = T_{p}(n)\), \(n \in \mathbb{N}_{0}\), \(p \in \mathbb{N}\).

For \(N \in \mathbb{N}\) with \(N \equiv 1~(\mathrm{mod}~2)\), by (1.3), we get
$$ \begin{aligned}[b] & 2 \sum _{k=0}^{N} (-1)^{k-1} e^{(k+x)t} \\ &\quad = \int _{\mathbb{Z}_{p}} e ^{(N+1+x+y)t} \,d\mu _{-1}(y) - \int _{\mathbb{Z}_{p}} e^{(x+y)t} \,d\mu _{-1}(y) \\ &\quad = \sum_{n=0}^{\infty } \biggl\{ \int _{\mathbb{Z}_{p}} (N+1+x+y)^{n} \,d\mu _{-1}(y) - \int _{\mathbb{Z}_{p}} (x+y)^{n} \,d\mu _{-1}(y) \biggr\} \frac{t ^{n}}{n!}. \end{aligned} $$
(2.14)

Now, we see that (2.14) is equivalent to the next theorem.

Theorem 2.6

For\(N \in \mathbb{N}\), with\(N \equiv 1~(\mathrm{mod}~2)\), and\(n \in \mathbb{N}_{0}\), we have
$$ T_{n}(N | x) = E_{n}(x+N+1) - E_{n}(x). $$
(2.15)
From (2.14), and recalling (1.13), we note that
$$ \begin{aligned}[b] 2 \sum _{k=0}^{N} (-1)^{k-1} e^{(k+x)t}& = e^{xt} \bigl(e^{(N+1)t} -1 \bigr) \int _{\mathbb{Z}_{p}} e^{yt} \,d\mu _{-1}(y) \\ & = \frac{2}{e^{t} +1} e^{xt} \Biggl( \sum _{m=0}^{N+1} \binom{N+1}{m} \bigl(e^{t}-1 \bigr)^{m} -1 \Biggr) \\ & = \Biggl(\sum_{j=0}^{\infty } E_{j}(x) \frac{t^{j}}{j!} \Biggr) \Biggl(\sum _{l=0}^{\infty } \Biggl(\sum_{m=0}^{l} \binom{N+1}{m} m! S_{2}(l,m) \Biggr) \frac{t^{l}}{l!} -1 \Biggr) \\ & = \sum_{n=0}^{\infty } \Biggl\{ \sum _{l=0}^{n} \sum_{m=0}^{l} \binom{N+1}{m} \binom{n}{l} m! S_{2}(l,m) E_{n-l}(x) - E_{n}(x) \Biggr\} \frac{t^{n}}{n!}, \end{aligned} $$
(2.16)
where \(N \in \mathbb{N}\) with \(N \equiv 1~(\mathrm{mod}~2)\) and \(S_{2}(l,m)\) is the Stirling number of the second kind.

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.7

For\(N \in \mathbb{N}\)with\(N \equiv 1~(\mathrm{mod}~2)\)and\(n \in \mathbb{N}_{0}\), we have
$$ T_{n}(N | x) = \sum_{l=0}^{n} \sum_{m=0}^{l} \binom{N+1}{m} \binom{n}{l} m! S_{2}(l,m) E_{n-l}(x) - E_{n}(x), $$
where\(S_{2}(n,m)\)is the Stirling number of the second kind.
For \(m,k \in \mathbb{N}\) with \(m - k \geq 1\), and making use of (1.2) and (2.9), we have
$$ \begin{aligned}[b] & (-1)^{m-k} \int _{\mathbb{Z}_{p}} x^{m-k} \,d\mu _{-1}(x) \\ &\quad = - \int _{\mathbb{Z}_{p}} x^{m-k} \,d\mu _{-1}(x) = \int _{\mathbb{Z}_{p}} (x+1)^{m-k} \,d\mu _{-1}(x) \\ & \quad = \sum_{j=0}^{m-k} \binom{m-k}{m-k-j} \int _{\mathbb{Z}_{p}} x^{m-k-j} \,d\mu _{-1}(x) = \sum _{j=k}^{m} \binom{m-k}{m-j} \int _{\mathbb{Z}_{p}} x ^{m-j} \,d\mu _{-1}(x) \\ &\quad = \frac{1}{\binom{m}{k}} \sum_{j=k}^{m} \binom{m}{j} \binom{j}{k} \int _{\mathbb{Z}_{p}} x^{m-j} \,d\mu _{-1}(x). \end{aligned} $$
(2.17)

Theorem 2.8

For\(m,k \in \mathbb{N}\)with\(m - k \geq 1\), we have
$$ (-1)^{m-k} \binom{m}{k} E_{m-k} = \sum _{j=k}^{m} \binom{m}{j} \binom{j}{k} E_{m-j}. $$

3 Some identities of degenerate Euler numbers and polynomials

In this section, we will introduce the degenerate alternating integer power sum polynomials and express them in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second, and derive some properties on degenerate Euler numbers and polynomials.

Throughout this section, we assume that \(\lambda \in \mathbb{C}_{p}\) with \(| \lambda |_{p} < p^{-\frac{1}{p-1}}\). The degenerate exponential function is defined as
$$ e_{\lambda }^{x} (t) = (1+ \lambda t)^{\frac{x}{\lambda }}, \qquad e _{\lambda } (t) = e_{\lambda }^{1} (t),\quad n \geq 0\mbox{ } (\text{see [3, 4, 14--16]}). $$

Note that \(\lim_{\lambda \rightarrow 0} e_{\lambda }^{x} (t) = e^{xt}\).

It is well known that the degenerate Euler polynomials are defined by L. Carlitz as
$$ \frac{2}{e_{\lambda } (t) +1} e_{\lambda }^{x} (t) = \frac{2}{(1+ \lambda t)^{\frac{1}{\lambda }} +1} (1+ \lambda t)^{\frac{x}{\lambda }} = \sum _{n=0}^{\infty } \mathcal{E}_{n,\lambda }(x) \frac{t^{n}}{n!}. $$
(3.1)
When \(x = 0\), \(\mathcal{E}_{n,\lambda } = \mathcal{E}_{n,\lambda }(0)\) are called the degenerate Euler numbers (see [3, 4, 14, 15, 16, 17]).
From (3.1), we note that
$$ \mathcal{E}_{n,\lambda }(x) = \sum _{l=0}^{n} \binom{n}{l} (x)_{n-l, \lambda } \mathcal{E}_{l,\lambda },\quad n \geq 0 \mbox{ } (\text{see [3, 4, 14--17]}), $$
(3.2)
where \((x)_{n,\lambda } = x (x-\lambda ) \cdots (x-(n-1)\lambda )\), \(n \geq 1\), \((x)_{0,\lambda } = 1\).
From (3.1), we can derive the following recurrence relation for \(\mathcal{E}_{n,\lambda }\), \(n \geq 0\).
$$ \begin{aligned}[b] \sum_{l=0}^{n} \binom{n}{l} (1)_{n-l,\lambda } \mathcal{E}_{l,\lambda } + \mathcal{E}_{n,\lambda } = \textstyle\begin{cases} 2, & \text{if } n = 0, \\ 0, & \text{if } n > 0. \end{cases}\displaystyle \end{aligned} $$
(3.3)
From (3.2) and (3.3), we have
$$ \mathcal{E}_{n,\lambda }(1) = - \mathcal{E}_{n,\lambda } + 2 \delta _{0,n},\quad n \geq 0. $$
For \(N \in \mathbb{N}\) with \(N \equiv 1~(\mathrm{mod}~2)\), we have
$$ \begin{aligned}[b] 2 \sum_{k=0}^{N} (-1)^{k-1} e_{\lambda }^{k+x}(t) & = \frac{2}{e_{ \lambda }(t) +1} \bigl( e_{\lambda }^{N+1+x}(t) - e_{\lambda }^{x}(t) \bigr) \\ & = \sum_{n=0}^{\infty } \bigl\{ \mathcal{E}_{n,\lambda }(N+1+x) - \mathcal{E}_{n,\lambda }(x) \bigr\} \frac{t^{n}}{n!}. \end{aligned} $$
(3.4)
On the other hand,
$$ 2 \sum_{k=0}^{N} (-1)^{k-1} e_{\lambda }^{k+x}(t) = \sum _{n=0}^{ \infty } \Biggl( 2 \sum _{k=0}^{N} (-1)^{k-1} (k+x)_{n,\lambda } \Biggr) \frac{t^{n}}{n!}. $$
(3.5)
Let us define a degenerate version of the alternating integer power sum polynomials, called the degenerate alternating integer power sum polynomials, by
$$ T_{p, \lambda } (n | x) = 2 \sum_{k=0}^{n} (-1)^{k-1} (k+x)_{p, \lambda },\quad n \geq 0. $$
(3.6)

Note that \(\lim_{\lambda \rightarrow 0} T_{p, \lambda } (n | x) = T _{p} (n | x)\), \(n \geq 0\).

Therefore, by (3.5) and (3.6), we obtain the following theorem.

Theorem 3.1

For\(n \in \mathbb{N}_{0}\), and\(N \in \mathbb{N}\), with\(N \equiv 1~( \mathrm{mod}~2)\), we have
$$ T_{n, \lambda } (N | x) = \mathcal{E}_{n,\lambda }(N+1+x) - \mathcal{E}_{n,\lambda }(x). $$
From (1.2), we note that
$$ \int _{\mathbb{Z}_{p}} e_{\lambda }^{x+y} (t) \,d\mu _{-1}(y) = \frac{2}{e _{\lambda } (t) +1} e_{\lambda }^{x} (t) = \sum _{n=0}^{\infty } \mathcal{E}_{n,\lambda }(x) \frac{t^{n}}{n!}. $$
(3.7)
On the other hand,
$$ \int _{\mathbb{Z}_{p}} e_{\lambda }^{x+y} (t) \,d\mu _{-1}(y) = \sum_{n=0} ^{\infty } \int _{\mathbb{Z}_{p}} (x+y)_{n,\lambda } \,d\mu _{-1}(y) \frac{t ^{n}}{n!}. $$
(3.8)
By (3.7) and (3.8), we get
$$ \int _{\mathbb{Z}_{p}} (x+y)_{n,\lambda } \,d\mu _{-1}(y) = \mathcal{E} _{n,\lambda }(x),\quad n \geq 0. $$
(3.9)
For \(d \in \mathbb{N}\) with \(d \equiv 1~(\mathrm{mod}~2)\), by (1.3), we get
$$ \begin{aligned}[b] & \int _{\mathbb{Z}_{p}} e_{\lambda }^{x+y} (t) \,d\mu _{-1}(y) \\ &\quad = \frac{2}{e _{\lambda }^{d} (t) +1} \sum_{l=0}^{d-1} (-1)^{l} e_{\lambda }^{x+l} (t) \\ &\quad = \frac{2}{e_{\frac{\lambda }{d}} (dt) +1} \sum_{l=0}^{d-1} (-1)^{l} e_{\frac{\lambda }{d}}^{\frac{x+l}{d}} (dt) = \sum _{l=0}^{d-1} (-1)^{l} \sum _{n=0}^{\infty } \mathcal{E}_{n,\frac{\lambda }{d}} \biggl( \frac{x+l}{d}\biggr) \frac{d^{n} t^{n}}{n!} \\ &\quad = \sum_{n=0}^{\infty } \Biggl(d^{n} \sum_{l=0}^{d-1} (-1)^{l} \mathcal{E}_{n,\frac{\lambda }{d}} \biggl(\frac{x+l}{d}\biggr) \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(3.10)
From (3.9) and (3.10), we have
$$ \mathcal{E}_{n,\lambda }(x) = d^{n} \sum _{l=0}^{d-1} (-1)^{l} \mathcal{E}_{n,\frac{\lambda }{d}} \biggl(\frac{x+l}{d}\biggr), $$
(3.11)
where \(n \in \mathbb{N}_{0}\) and \(d \in \mathbb{N}\) with \(d \equiv 1~( \mathrm{mod}~2)\).
From (3.7) and (3.9), we have
$$ \int _{\mathbb{Z}_{p}} (1-x+y)_{n,\lambda } \,d\mu _{-1}(y) = (-1)^{n} \int _{\mathbb{Z}_{p}} (x+y)_{n,-\lambda } \,d\mu _{-1}(y), $$
(3.12)
where n is a nonnegative integer.
Hence, by (3.7), we get
$$ \mathcal{E}_{n,\lambda }(1-x) = (-1)^{n} \mathcal{E}_{n,-\lambda }(x), \quad n \geq 0. $$
(3.13)
Now, we observe that
$$ \begin{aligned}[b] 2 \sum_{k=0}^{N} (-1)^{k-1} e_{\lambda }^{x+k}(t) & = \frac{2}{e_{ \lambda }(t) +1} \bigl( e_{\lambda }^{N+1+x}(t) - e_{\lambda }^{x}(t) \bigr) \\ & = \frac{2}{e_{\lambda }(t) +1} \bigl( \bigl(e_{\lambda }(t)-1+1\bigr)^{N+1} - 1 \bigr) e_{\lambda }^{x}(t) \\ & = \frac{2}{e_{\lambda }(t) +1} \Biggl(\sum_{m=0}^{N+1} \binom{N+1}{m} \bigl(e_{\lambda }(t)-1\bigr)^{m}-1 \Biggr) e_{\lambda }^{x}(t), \end{aligned} $$
(3.14)
where \(N \in \mathbb{N}\), with \(N \equiv 1~(\mathrm{mod}~2)\).
As is well known, the degenerate Stirling numbers of the second kind are given by the generating function as
$$ \frac{1}{k!} \bigl(e_{\lambda }(t) -1 \bigr)^{k} = \sum_{n=k}^{\infty } S_{2, \lambda }(n,k) \frac{t^{n}}{n!}\quad (\text{see [14, 16, 18]}). $$
(3.15)
From (3.14) and (3.15), we have
$$ \begin{aligned}[b] 2 \sum_{k=0}^{N} (-1)^{k-1} e_{\lambda }^{x+k}(t) & = \sum _{j=0}^{ \infty } \mathcal{E}_{j,\lambda }(x) \frac{t^{j}}{j!} \Biggl(\sum_{l=0} ^{\infty } \sum_{m=0}^{l} \binom{N+1}{m} m! S_{2,\lambda }(l,m) \frac{t ^{l}}{l!}-1 \Biggr) \\ & = \sum_{n=1}^{\infty } \Biggl(\sum _{l=1}^{n} \sum_{m=1}^{l} \binom{n}{l} \binom{N+1}{m} m! \mathcal{E}_{n-l,\lambda }(x) S_{2, \lambda }(l,m) \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$
(3.16)
The left-hand side of (3.16) is given by
$$ \begin{aligned}[b] 2 \sum_{k=0}^{N} (-1)^{k-1} e_{\lambda }^{x+k}(t) & = \sum _{n=0}^{ \infty } \Biggl(2 \sum _{k=0}^{N} (-1)^{k-1} (x+k)_{n,\lambda } \Biggr) \frac{t ^{n}}{n!}. \\ & = \sum_{n=1}^{\infty } T_{n, \lambda } (N | x) \frac{t^{n}}{n!}, \end{aligned} $$
(3.17)
where \(N \in \mathbb{N}\) with \(N \equiv 1~(\mathrm{mod}~2)\).

Therefore, by (3.16) and (3.17), we obtain the following theorem.

Theorem 3.2

For\(n, N \in \mathbb{N}\), with\(N \equiv 1~(\mathrm{mod}~2)\), we have
$$ T_{n, \lambda } (N | x) = \sum_{l=1}^{n} \sum_{m=1}^{l} \binom{n}{l} \binom{N+1}{m} m! \mathcal{E}_{n-l,\lambda }(x) S_{2,\lambda }(l,m). $$

4 Conclusions

As is well known, the alternating integer power sums can be expressed in terms of some values of Euler polynomials. In this paper, we studied some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials, which are derived from certain fermionic p-adic integrals on \(\mathbb{Z}_{p}\). Here we mention that fermionic p-adic integrals were introduced by Kim and have been used fruitfully in investigations of combinatorial and number-theoretic aspects of many special numbers and polynomials.

Specifically, we obtained a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and also of Euler polynomials together with Stirling numbers of the second kind. Along the way, various properties of Euler numbers and polynomials were derived as well. As to the degenerate alternating integer power sum polynomials associated with the alternating integer power sums, we obtained their representations in terms of degenerate Euler polynomials and also of degenerate Euler polynomials together with the degenerate Stirling numbers of the second kind. Along the way, we also derived some properties of degenerate Euler numbers and polynomials.

Notes

Acknowledgements

We would like to thank the referees for their valuable comments and suggestions that improved the original manuscript in its present form.

Authors’ contributions

Each of the authors TK, DSK, HYK, and JK contributed to each part of this study equally, as well as read and approved the final version of the manuscript.

Funding

This research received no external funding.

Competing interests

The authors declare that they have no competing interests.

References

  1. 1.
    Araci, S., Acikgoz, M.: A note on the Frobenius–Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 22(3), 399–406 (2012) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bayad, A., Chikhi, J.: Apostol–Euler polynomials and asymptotics for negative binomial reciprocals. Adv. Stud. Contemp. Math. (Kyungshang) 24(1), 33–37 (2014) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Carlitz, L.: A degenerate Staudt–Clausen theorem. Arch. Math. (Basel) 7, 28–33 (1956) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51–88 (1979) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dolgy, D.V., Kim, D.S., Kwon, J., Kim, T.: Some identities of ordinary and degenerate Bernoulli numbers and polynomials. Symmetry 11(7), 847 (2019) CrossRefGoogle Scholar
  6. 6.
    Kim, D.S.: Identities associated with generalized twisted Euler polynomials twisted by ramified roots of unity. Adv. Stud. Contemp. Math. (Kyungshang) 22(3), 363–377 (2012) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kim, D.S., Dolgy, D.V., Kim, T., Kim, D.: Extended degenerate r-central factorial numbers of the second kind and extended degenerate r-central Bell polynomials. Symmetry 11(4), 595 (2019) CrossRefGoogle Scholar
  8. 8.
    Kim, D.S., Dolgy, D.V., Kwon, J., Kim, T.: Note on type 2 degenerate q-Bernoulli polynomials. Symmetry 11(7), 914 (2019) CrossRefGoogle Scholar
  9. 9.
    Kim, D.S., Kim, H.Y., Kim, D., Kim, T.: Identities of symmetry for type 2 Bernoulli and Euler polynomials. Symmetry 11(5), 613 (2019) CrossRefGoogle Scholar
  10. 10.
    Kim, D.S., Lee, N., Na, J., Park, K.H.: Identities of symmetry for higher-order Euler polynomials in three variables (I). Adv. Stud. Contemp. Math. (Kyungshang) 22(1), 51–74 (2012) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kim, T.: q-Euler numbers and polynomials associated with p-adic q-integrals. J. Nonlinear Math. Phys. 14(1), 15–27 (2007) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kim, T.: New approach to q-Euler polynomials of higher order. Russ. J. Math. Phys. 17(2), 218–225 (2010) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kim, T., Kim, D.S.: Degenerate Laplace transform and degenerate gamma function. Russ. J. Math. Phys. 24(2), 241–248 (2017) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kim, T., Kim, D.S.: Degenerate Bernstein polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2913–2920 (2019) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kim, T., Kim, D.S.: A note on type 2 Changhee and Daehee polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2763–2771 (2019) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kim, T., Kim, D.S.: Degenerate central Bell numbers and polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2507–2513 (2019) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kim, T., Kim, D.S., Jang, G.-W., Kwon, J.: A note on degenerate Bernstein polynomials. J. Inequal. Appl. 2019, 129, 12 pp. (2019) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kim, W.J., Kim, D.S., Kim, H.Y., Kim, T.: Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomials. J. Inequal. Appl. 2019, 160, 11 pp. (2019) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rim, S.-H., Jeong, J.: On the modified q-Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. (Kyungshang) 22(1), 93–98 (2012) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sen, E.: Theorems on Apostol–Euler polynomials of higher order arising from Euler basis. Adv. Stud. Contemp. Math. (Kyungshang) 23(2), 337–345 (2013) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Simsek, Y.: Identities and relations related to combinatorial numbers and polynomials. Proc. Jangjeon Math. Soc. 20(1), 127–135 (2017) MathSciNetzbMATHGoogle Scholar

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© The Author(s) 2019

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 and Affiliations

  • Taekyun Kim
    • 1
    • 2
  • Dae San Kim
    • 3
  • Han Young Kim
    • 2
  • Jongkyum Kwon
    • 4
    Email author
  1. 1.School of ScienceXian Technological UniversityXianChina
  2. 2.Department of MathematicsKwangwoon UniversitySeoulRepublic of Korea
  3. 3.Department of MathematicsSogang UniversitySeoulRepublic of Korea
  4. 4.Department of Mathematics Education and ERIGyeongsang National UniversityJinjuRepublic of Korea

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