On the Structure of Hyperexponential-Hypergeometric Functions and the Termination of Creative Telescoping

The method of creative telescoping, first introduced by Zeilberger in the 1990's, plays an important role in the study of special functions. In particular, it is applicable to the verification of a large class of identities that involve integrals or sums of proper hyperexponential-hypergeometric functions. In this talk, I shall present a structure theorem for multivariate hyperexponential-hypergeometric functions that generalizes both Abramov and Petkovsek's result for multivariate hypergeometric terms and the recent result by Feng, Singer, and Wu for bivariate hyperexponential-hypergeometric functions. As an application of this structure theorem, I shall then present two criteria for the existence of telescopers for bivariate hyperexponential-hypergeometric functions: one for telescopers with respect to the continuous variable, the other for telescopers with respect to the discrete one. Those criteria decide the termination of Zeilberger's algorithm in the bivariate continuous-discrete setting. Work in progress with F. Chyzak, R. Feng, and Z. Li.