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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2023

Values of E-functions are not Liouville numbers

Stéphane Fischler
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Tanguy Rivoal

Résumé

Shidlovskii has given a linear independence measure of values of $E$-functions with rational Taylor coefficients at a rational point, not a singularity of the underlying differential system satisfied by these $E$-functions. Recently, Beukers has proved a qualitative linear independence theorem for the values at an algebraic point of $E$-functions with arbitrary algebraic Taylor coefficients. In this paper, we obtain an analogue of Shidlovskii's measure for values of arbitrary $E$-functions at algebraic points. This enables us to solve a long standing problem by proving that the value of an $E$-function at an algebraic point is never a Liouville number. We also prove that values at rational points of $E$-functions with rational Taylor coefficients are linearly independent over $\overline{\mathbb{Q}}$ if and only if they are linearly independent over $\mathbb{Q}$. Our methods rest upon improvements of results obtained by Andr\'e and Beukers in the theory of $E$-operators.
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Dates et versions

hal-03920532 , version 1 (03-01-2023)
hal-03920532 , version 2 (24-01-2023)
hal-03920532 , version 3 (06-12-2023)

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Stéphane Fischler, Tanguy Rivoal. Values of E-functions are not Liouville numbers. 2023. ⟨hal-03920532v3⟩
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