On Siegel’s problem for E-functions

Stéphane Fischler; Tanguy Rivoal

Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 83-115.

Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. He asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of E-functions of the form p F q (λz q-p+1 ), q≥p≥1, with rational parameters. The case of E-functions of differential order less than or equal to 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring 𝐆 of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively “small” ring 𝐇 generated by algebraic numbers, 1/π and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of 𝐆 is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove (in two steps) that the coefficients of the asymptotic expansion of a hypergeometric E-function with rational parameters are in 𝐇. Finally, we prove a similar result for G-functions.

Classification : 33, 11, 41