Radial behavior of Mahler functions.

Many papers have been recently devoted to the study of the radial behavior as z → 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) ρ for some ρ ∈ ℂ and C (z) is a real analytic function of z ∈ (0 , 1) such that C (z) = C (z r). They did not provide a formula for C (z) which was made explicit only in a few examples of r-Mahler functions of orders 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 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 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. [ABSTRACT FROM AUTHOR]