Computer Algebra and Power Series with Positive Coecien ts
Manuel Kauers · 2007
We consider the question whether all the coecien ts in the series expansions of some specic rational functions are positive, and we demonstrate how computer algebra can help answering questions arising in this context. By giving partial computer proofs, we provide new evidence in support of some longstanding open conjectures. Also two new conjectures are made. Proving that all the coecien ts in the series expansion of some given multivariate rational function are positive can be quite a dicult task. There are dicult papers on this subject by Szego (14), Askey and Gasper (2), Koornwinder (10), and others. Gillis, Reznick and Zeilberger (8) have pointed out that some seemingly dicult positivity results can be proven also by elementary means. In this paper, we make an attempt at going one step further: We ask to which extent positivity results can be proven automatically using computer algebra. Two results from the literature and two longstanding open conjectures related to them are considered. For none of the latter we are able to provide full proofs, but we give partial proofs that add new evidence in support of these conjectures.