The Abel-Zeilberger Algorithm

TL;DR

This paper introduces the Abel-Zeilberger algorithm, combining Abel's lemma and Zeilberger's algorithm to find recurrence relations and verify identities in summations involving special numbers.

Contribution

It extends Abel's lemma to linear difference operators and applies the combined algorithm to prove and discover identities involving harmonic, derangement, Fibonacci numbers, and more.

Findings

Successfully proves several classical identities.

Extends Abel's lemma to polynomial coefficient difference operators.

Provides a new algorithmic approach for summation identities.

Abstract

We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.