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Preprints, Working Papers, ... Year : 2022

Radial behavior of Mahler functions

Marina Poulet
  • Function : Author
Tanguy Rivoal

Abstract

Many papers have been recently devoted to the study of the radial behavior as $z\to 1^-$ of transcendental $r$-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, $r$-Mahler functions behave like $(1+o(1))C(z)/(1-z)^\rho$ for some $\rho\in \mathbb C$ and $C(z)$ is a real analytic function of $z\in [0,1]$ such that $C(z)=C(z^r)$. They did not provide any formula for $C(z)$ which has been explicited only in a few examples of $r$-Mahler functions of order~1 and~2, and for specific values of~$r$. In this paper, we first provide an explicit expression of $C(z)$ as an exponential of a Fourier series in the variable $\log\log(1/z)/\log(r)$ for every $r$-Mahler function of order~1. Then, extending to a large setting a method introduced by Brent-Coons-Zudilin in 2016 to compute $C(z)$ associated to the Dilcher-Stolarsky function (a $4$-Mahler function of order~2 in $\mathbb Q[[z]]$), we provide an explicit expression of $C(z)$ for every $r$-Mahler function of order~2 under mild assumptions on the coefficients in $\mathbb R(z)$ of the underlying $r$-Mahler equations. This applies in particular to the generating function of the Baum-Sweet sequence. We do the same for $r$-Mahler functions solutions of inhomogeneous Mahler equations of order~1 and we conclude the paper with possible generalizations to $r$-Mahler equations of order~$\ge 3$.
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Dates and versions

hal-03703010 , version 1 (23-06-2022)

Identifiers

  • HAL Id : hal-03703010 , version 1

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Marina Poulet, Tanguy Rivoal. Radial behavior of Mahler functions. 2022. ⟨hal-03703010⟩
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