Shaoshi Chen , Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences

In 1992, Wilf and Zeilberger conjectured that a hypergeometric term in several discrete and continuous variables is holonomic if and only if it is proper. Strictly speaking the conjecture does not hold, but it is true when reformulated properly: Payne proved a piecewise interpretation in 1997, and independently, Abramov and Petkovsek in 2002 proved a conjugate interpretation. Both results address the pure discrete case of the conjecture. In this paper we extend their work to hypergeometric terms in several discrete and continuous variables and prove the conjugate interpretation of the Wilf-Zeilberger conjecture in this mixed setting. With the proof of this conjecture, one now could algorithmically detect the holonomicity of hypergeometric terms by checking properness with the algorithms in the work by Chen et al. This is important because it gives a simple test for the termination of Zeilberger's algorithm. This is joint work with Christoph Koutschan (Austrian Academy of Sciences).