The Abel-Zeilberger Algorithm

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 Apéry-Schmidt-Strehl identity, Calkin's identity and some identities involving Fibonacci numbers.

The expected value under the Yule model of the squared path-difference distance

“…2]. We shall prove (5) and (6) using the technique introduced in [3]. The main ingredient is Abel's lemma on summation by parts: for every two sequences (a k ) k and (b k ) k ,…”

A note on Abel’s partial summation formula

“…Abel's partial summation formula (also known as Abel's transformation) asserts that every pair of families (a k ) n k=1 and (b k ) n k=1 of complex numbers verifies the identity (Ab ↑ ) This identity, that appears in the proof of Theorem III in [1], is instrumental in deriving a number of important results such as the Abel-Dirichlet criterion of convergence for signed series, the Abel theorem on power series, the Abel summation method (see [4], [24]), Kronecker's lemma about the relationship between convergence of infinite sums and convergence of sequences (see [21], Lemma IV.3.2, p. 390), algorithms for establishing identities involving harmonic numbers and derangement numbers [3], the variational characterization of the level sets corresponding to majorization in R N [25], Mertens' proof of his theorem on the sum of the reciprocals of the primes [26] etc.…”

Chu–Vandermonde convolution and harmonic number identities

“…For l ∈ N and n ∈ N 0 , define the generalized harmonic numbers by There exist many elegant identities involving harmonic numbers. They can be found in the papers [2], [3], [4], [5], [6], [7], [8], [9] and [10].…”