Values of globally bounded 𝐺-functions

Fischler, S.; Rivoal, T.

doi:10.1090/proc/14402

Values of globally bounded $G$-functions
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by S. Fischler and T. Rivoal PDF

Proc. Amer. Math. Soc. 147 (2019), 2321-2330 Request permission

Abstract:

In this paper we define and study a filtration $(\mathbf {G}_s)_{s\geq 0}$ on the algebra of values at algebraic points of analytic continuations of $G$-functions: $\mathbf {G}_s$ is the set of values at algebraic points in the disk of convergence of all $G$-functions $\sum _{n=0}^\infty a_n z^n$ for which there exist some positive integers $b$ and $c$ such that $d_{bn}^{s} c^{n+1} a_n$ is an algebraic integer for any $n$, where $d_n =\text {lcm}(1,2,\ldots ,n)$.

We study the situation at the boundary of the disk of convergence, and using transfer results from analysis of singularities we deduce that constants in $\mathbf {G}_s$ appear in the asymptotic expansion of such a sequence $(a_n)$.

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