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.
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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
From (1.1), we note that
and by induction, for any \(n \in \mathbb{N}\), we get
It is well known that the Euler polynomials are defined by
When \(x=0\), \(E_{n} = E_{n}(0)\) are called the Euler numbers.
From (1.4), we note that
where n is a nonnegative integer.
Let
Then, by (1.4) and (1.5), we get
where \(n \in \mathbb{N}\), with \(n \equiv 1~(\mathrm{mod}~ 2)\). Thus we have, for \(n,p \in \mathbb{N}\), with \(n \equiv 1~(\mathrm{mod}~2)\),
From (1.2), we can derive the following equation (1.10):
Thus, by (1.10), we get
Thus, by (1.9) and (1.11), we have
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
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
From (2.1), for \(n,p \in \mathbb{N}\) with \(n \equiv 1~( \mathrm{mod}~ 2)\), we note that
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
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
For \(n \in \mathbb{N}_{0} = \mathbb{N} \cup \{0\}\), we have
Thus, by (2.6), we get
For \(n \in \mathbb{N}_{0}\), and by (1.2), we have
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
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
For the next result, we note that, for any \(n \in \mathbb{N}\),
For \(m,n \in \mathbb{N}\), and by (1.11) and (2.9), we have
On the other hand, by (2.6) and (2.8), we get
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:
Now, we define the alternating integer power sum polynomials by
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
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
From (2.14), and recalling (1.13), we note that
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
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
Theorem 2.8
For \(m,k \in \mathbb{N}\) with \(m - k \geq 1\), we have
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
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
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
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 \(\mathcal{E}_{n,\lambda }\), \(n \geq 0\).
For \(N \in \mathbb{N}\) with \(N \equiv 1~(\mathrm{mod}~2)\), we have
On the other hand,
Let us define a degenerate version of the alternating integer power sum polynomials, called the degenerate alternating integer power sum polynomials, by
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
From (1.2), we note that
On the other hand,
For \(d \in \mathbb{N}\) with \(d \equiv 1~(\mathrm{mod}~2)\), by (1.3), we get
From (3.9) and (3.10), we have
where \(n \in \mathbb{N}_{0}\) and \(d \in \mathbb{N}\) with \(d \equiv 1~( \mathrm{mod}~2)\).
where n is a nonnegative integer.
Hence, by (3.7), we get
Now, we observe that
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
From (3.14) and (3.15), we have
The left-hand side of (3.16) is given by
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
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.
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Kim, T., Kim, D.S., Kim, H.Y. et al. Ordinary and degenerate Euler numbers and polynomials. J Inequal Appl 2019, 265 (2019). https://doi.org/10.1186/s13660-019-2221-5
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DOI: https://doi.org/10.1186/s13660-019-2221-5