Hadamard products of algebraic functions

JOURNAL OF NUMBER THEORY

卷:145

Article

Rivoal, T.¹  Roques, J.¹ 

[1] Univ Grenoble 1, CNRS UMR 5582, Inst Fourier, F-38402 St Martin Dheres, France

关键词: Hadamard products;    Hypergeometric series;    Rohrlich-Lang Conjecture;    Taylor coefficients of algebraic functions;   

DOI  :  10.1016/j.jnt.2014.06.015

来源: Elsevier

【 摘 要 】

Allouche and Mendes France [1] have defined the grade of a formal power series with algebraic coefficients as the smallest integer k such that this series is the Hadamard product of k algebraic power series. In this paper, we obtain lower and upper bounds for the grade of hypergeometric series by comparing two different asymptotic expansions of their Taylor coefficients, one obtained from their definition and another one obtained when assuming that the grade has a certain value. In such expansions, Gamma values at rational points naturally appear and our results mostly depend on the Rohrlich-Lang Conjecture for polynomial relations in Gamma values. We also obtain unconditional and sharp results when we can apply Diophantine results such as the Wolfart-Wustholz Theorem for Beta values. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free