Linear independence of values of -functions

  • Stéphane Fischler

    Université Paris-Saclay, Orsay, France
  • Tanguy Rivoal

    Université Grenoble Alpes, Grenoble, France
Linear independence of values of $G$-functions cover

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Given any non-polynomial -function of radius of convergence , we consider the -functions for any integers and . For any fixed algebraic number such that and any number field containing and the 's, we define as the -vector space generated by the values , and . We prove that for any , with effective constants and , and that the family contains infinitely many irrational numbers. This theorem applies in particular when is a hypergeometric series with rational parameters or a multiple polylogarithm, and it encompasses a previous result by the second author and Marcovecchio in the case of polylogarithms. The proof relies on an explicit construction of Padé-type approximants. It makes use of results of André, Chudnovsky and Katz on -operators, of a new linear independence criterion à la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.

Cite this article

Stéphane Fischler, Tanguy Rivoal, Linear independence of values of -functions. J. Eur. Math. Soc. 22 (2020), no. 5, pp. 1531–1576

DOI 10.4171/JEMS/950