Radial behavior of Mahler functions

Abstract

Many papers have been recently devoted to the study of the radial behavior as [Formula: see text] of transcendental [Formula: see text]-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, [Formula: see text]-Mahler functions behave like [Formula: see text] for some [Formula: see text] and [Formula: see text] is a real analytic function of [Formula: see text] such that [Formula: see text]. They did not provide a formula for [Formula: see text] which was made explicit only in a few examples of [Formula: see text]-Mahler functions of orders 1 and 2, and for specific values of [Formula: see text]. In this paper, we first provide an explicit expression of [Formula: see text] as an exponential of a Fourier series in the variable [Formula: see text] for every [Formula: see text]-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute [Formula: see text] associated to the Dilcher–Stolarsky function (a [Formula: see text]-Mahler function of order 2 in [Formula: see text]), we provide an explicit expression of [Formula: see text] for every [Formula: see text]-Mahler function of order 2 under mild assumptions on the coefficients in [Formula: see text] of the underlying [Formula: see text]-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for [Formula: see text]-Mahler functions solutions of inhomogeneous Mahler equations of order 1.