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Problème de Riemann et irrationalité d’un quotient de deux fonctions hypergéométriques de Gauss. (On a problem of Riemann and the irrationality of the ratio of two Gaussian hypergeometric functions). (French) Zbl 0607.10026
The author explicitly constructs Padé approximants to the quotient of two continuous hypergeometric functions, and deduces sharp measures of irrationality for its values at certain rational points from the arithmetic properties of binomial coefficients. The argument follows a method of D. V. and G. V. Chudnovsky [Lect. Notes Math. 1052, 37-84, 85-167 (1984; Zbl 0536.10028, Zbl 0536.10029)] based on the knowledge of the monodromy group of hypergeometric equations, and on Fuchs’ relation on exponents - it is the latter, rather than linear algebra, which provides the exact order at 0 of the remainder of the Padé approximation.
Reviewer: D.Bertrand

11J81 Transcendence (general theory)
41A21 Padé approximation
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)