General Convolution Identities for Bernoulli and Euler Polynomials

TL;DR

This paper derives broad convolution identities for Bernoulli and Euler polynomials using difference operators, symbolic computation, and probability theory, unifying and extending many known identities including those of Miki and Matiyasevich.

Contribution

It introduces a general framework for convolution identities of Bernoulli and Euler polynomials that encompasses many existing results as special cases.

Findings

Derived general kth order convolution identities for Bernoulli and Euler polynomials.

Unified known identities such as Miki and Matiyasevich within a single formula.

Identified parameters that generate numerous new identities for these polynomials.

Abstract

Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general kth order (k \ge 2) convolution identities for Bernoulli and Euler polynomials. This is achieved by use of an elementary result on uniformly distributed random variables. These identities depend on k positive real parameters, and as special cases we obtain numerous known and new identities for these polynomials. In particular we show that the well-known identities of Miki and Matiyasevich for Bernoulli numbers are special cases of the same general formula.