Convolution Identities for Bernoulli and Genocchi Polynomials

Takashi Agoh

doi:10.37236/3489

Abstract

The main purpose of this paper is to derive various Matiyasevich-Miki-Gessel type convolution identities for Bernoulli and Genocchi polynomials and numbers by applying some Euler type identities with two parameters.

Highlights

The Bernoulli numbers Bn and polynomials Bn(x) appear in many areas of mathematics and theoretical physics, most notably in number theory, the calculus of finite differences, asymptotic analysis and quantum field theory
Concerning convolution identities for Bernoulli numbers, the most basic and remarkable one is the following formula, which is usually attributed to Euler: nn i BiBn−i = −nBn−1 − (n − 1)Bn (n 1)
This identity was extended to Bernoulli polynomials and generalized in many directions

Summary

Introduction

The Bernoulli numbers Bn and polynomials Bn(x) appear in many areas of mathematics and theoretical physics, most notably in number theory, the calculus of finite differences, asymptotic analysis and quantum field theory They can be defined by the generating functions t. Concerning convolution identities for Bernoulli numbers, the most basic and remarkable one is the following formula, which is usually attributed to Euler: nn i BiBn−i = −nBn−1 − (n − 1)Bn (n 1). This identity was extended to Bernoulli polynomials and generalized in many directions (see, e.g., [1, 2, 3, 4, 5, 7, 13]). This is essentially the same idea used by Crabb in [6] to give another short and intelligible proof of (1.7)

Euler type convolution identities

M-M-G type convolution identities

Additional remarks