  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.
