Positivity of rational functions and their diagonals

JOURNAL OF APPROXIMATION THEORY

卷:195

Article

Straub, Armin¹  Zudilin, Wadim² 

[1] Max Planck Inst Math, D-53111 Bonn, Germany

[2] Univ Newcastle, Sch Math & Phys Sci, Callaghan, NSW 2308, Australia

关键词: Positivity;    Rational function;    Hypergeometric function;    Modular function;    Apery-like sequence;    Multivariate asymptotics;   

DOI  :  10.1016/j.jat.2014.05.012

来源: Elsevier

【 摘 要 】

The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szego as well as Askey and Gasper, who inspired more recent work. It is well known that the diagonal coefficients of rational functions are D-finite. This note is motivated by the observation that, for several of the rational functions whose positivity has received special attention, the diagonal terms in fact have arithmetic significance and arise from differential equations that have modular parametrization. In each of these cases, this allows us to conclude that the diagonal is positive. Further inspired by a result of Gillis, Reznick and Zeilberger, we investigate the relation between positivity of a rational function and the positivity of its diagonal. Crown Copyright (C) 2014 Published by Elsevier Inc. All rights reserved.

【 授权许可】

Free