On the values of GG-functions

  • Stéphane Fischler

    Université Paris-Sud 11, Orsay, France
  • Tanguy Rivoal

    Université Grenoble I, Saint-Martin-D'hères Cedex, France


In this paper we study the set G\mathbf{G} of values at algebraic points of analytic continuations of GG-functions (in the sense of Siegel). This subring of C\mathbb{C} contains values of elliptic integrals, multiple zeta values, and values at algebraic points of generalized hypergeometric functions p+1Fp_{p+1}F_p with rational coefficients. Its group of units contains non-zero algebraic numbers, π\pi, Γ(a/b)b\Gamma(a/b)^b and B(x,y)B(x,y) (with a,bZa,b\in \mathbb{Z} such that a/b∉Za/b\not\in \mathbb Z, and x,yQx,y\in\mathbb{Q} such that B(x,y)B(x,y) exists and is non-zero). We prove that for any ξG\xi \in \mathbf{G}, both Reξ\operatorname{Re} \xi and Imξ\mathrm{Im} \, \xi can be written as f(1)f(1), where ff is a GG-function with rational coefficients of which the radius of convergence can be made arbitrarily large. As an application, we prove that quotients of elements of GR\mathbf{G} \cap \mathbb{R} are exactly the numbers which can be written as limits of sequences an/bna_n/b_n, where n=0anzn\sum_{n=0}^{\infty} a_n z^n and n=0bnzn\sum_{n=0}^{\infty} b_n z^n are GG-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for ζ(3)\zeta(3), and gives answers to questions asked by T. Rivoal in “Approximations rationnelles des valeurs de la fonction Gamma aux rationnels: le cas des puissances”, Acta Arith. 142 (2010), no. 4, 347–365.

Cite this article

Stéphane Fischler, Tanguy Rivoal, On the values of GG-functions. Comment. Math. Helv. 89 (2014), no. 2, pp. 313–341

DOI 10.4171/CMH/321