Extended Zeilberger's Algorithm for Identities on Bernoulli and Euler Polynomials

TL;DR

This paper introduces an extended Zeilberger's algorithm to prove identities involving Bernoulli and Euler polynomials by leveraging contour integral definitions and recurrence relations, simplifying the proof process.

Contribution

It extends Zeilberger's algorithm to handle Bernoulli and Euler polynomial identities using contour integrals and parameter-free recurrence relations.

Findings

Successfully proved several identities on Bernoulli and Euler polynomials.

Demonstrated the effectiveness of the extended algorithm in symbolic computation.

Simplified proofs by avoiding direct integral computation.

Abstract

We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals.