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

2021/06/02 by Su Hu, Min-Soo Kim, Hu, Su +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2106.01119

Abstract

By using Cauchy's formula, it is known that Bernoulli numbers and Euler numbers can be represented by the contour integrals \beginaligned B_namp;=(n!)/(2πi)\oint (z)/(e^z-1)\fracd zzⁿ⁺¹,
E_namp;=(n!)/(2πi)\oint \frac2e^ze^2z+1\fracd zzⁿ⁺¹, \endaligned while the following Witt's formula represents Euler polynomials through the fermionic p-adic integrals E_n(a)=∫_{\mathbb Zp}(x+a)ⁿdμ_-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.

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