HADAMARD PRODUCTS OF ALGEBRAIC FUNCTIONS

Abstract. Allouche and Mendès-France (in Hadamard grade of power series, J. Num-ber Theory 131 (2011)) 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 poly-nomial relations in Gamma values. We also obtain unconditional and sharp results when we can apply Diophantine results such as the Wolfart-Wüstholz Theorem for Beta values. 1