Identities for the Euler polynomials, $p$-adic integrals and Witt's formula

TL;DR

This paper derives new identities and symmetry properties for Euler polynomials using contour integrals, p-adic integrals, and Witt's formula, expanding the understanding of their algebraic structure.

Contribution

It introduces novel identities and symmetry properties for Euler polynomials based on p-adic integrals and contour integral representations.

Findings

Discovered new symmetry properties of Euler polynomials.

Derived identities including Kaneko-Momiyama and Alzer-Kwong types.

Extended the algebraic understanding of Euler polynomials.

Abstract

By using Cauchy's formula, it is known that Bernoulli numbers and Euler numbers can be represented by the contour integrals \begin{equation*} \begin{aligned} B_n&=\frac{n!}{2\pi i}\oint \frac{z}{e^z-1}\frac{d z}{z^{n+1}},\label{condefi}\\[4pt] E_n&=\frac{n!}{2\pi i}\oint \frac{2e^z}{e^{2z}+1}\frac{d z}{z^{n+1}}, \end{aligned} \end{equation*} while the following Witt's formula represents Euler polynomials through the fermionic $p$-adic integrals $$E_n(a)=\int_{\mathbb Z_p}(x+a)^nd\mu_{-1}(x).$$ Base on the above Witt's identity and the binomial theorem, we prove some new identities for the Euler polynomials briefly. In particular, some symmetry properties of Euler polynomials have been discovered, which implies many interesting identities (known or unknown), including the Kaneko-Momiyama type identities (shown by Wu, Sun, and Pan) and the Alzer-Kwong type identity for Euler polynomials.