Identities of symmetry for Bernoulli polynomials and power sums

Taekyun Kim,Dae San Kim,Jongkyum Kwon,Han Young Kim

doi:10.1186/s13660-020-02511-9

Abstract

Identities of symmetry in two variables for Bernoulli polynomials and power sums had been investigated by considering suitable symmetric identities. T. Kim used a completely different tool, namely the p-adic Volkenborn integrals, to find the same identities of symmetry in two variables. Not much later, it was observed that this p-adic approach can be generalized to the case of three variables and shown that it gives some new identities of symmetry even in the case of two variables upon specializing one of the three variables. In this paper, we generalize the results in three variables to those in an arbitrary number of variables in a suitable setting and illustrate our results with some examples.

Highlights

Introduction and preliminariesTuenter [17] obtained the following identity of symmetry involving the Bernoulli numbers and the power sums
The possible numbers of distinct identities of symmetry are 1, 2, 3, and 6 corresponding to the quotient |S3|/|H|, where H is a subgroup of S3, with the respective orders 6, 3, 2, and 1
4 Conclusion Identities of symmetry in two variables for Bernoulli polynomials and power sums, which had been shown by using suitable symmetric identities, were derived by employing a completely different tool in [12], namely the p-adic Volkenborn integrals

Summary

Introduction

Tuenter [17] obtained the following identity of symmetry involving the Bernoulli numbers and the power sums. This was done by showing that the exponential generating function of the sum on the left-hand side of (1) is invariant under the interchange of w1 and w2. Which was proved by Deeba and Rodriguez [2] and Gessel [3]. It was a conjecture posed by Namias who found identity (2) for w1 = 2, 3 by using the multiplication formula for the gamma function (see [2])

Objectives

Conclusion