Terminating 3F2(2) and Some Identities for 2F2

The Petkovsek-Wilf-Zeilberger’s algorithm permits to give a simple proof of the Rathie-Pogany’s identity for the hypergeometric function 2F2.Journal of Institute of Science and Technology, 2015, 20(1): 123-126

On Gessel-Kaneko's identity for Bernoulli numbers

The present work deals with Bernoulli numbers. Using Zeilberger's algorithm, we generalize an identity on Bernoulli numbers of Gessel-Kaneko's type. Appendix written by Ira M. Gessel offers a closely related formula via umbral calculus.

The Abel-Zeilberger Algorithm

By combining Abel's lemma on summation by parts with Zeilberger's algorithm, we give an algorithm, called the Abel-Zeilberger 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, an identity of Andrews and Paule, and an identity of Calkin.

Summation identities for representation of certain real numbers

We present identities used to represent real numbers of the formxum±yvnfor appropriately chosen real numbersx,y,u,vand nonnegative integersmandn. We present the proofs of the identities by applying Zeilberger's algorithm.