Algebraic independence of the generating functions of Stern's sequence and of its twist

Abstract

Very recently, the generating function A(z) of the Stern sequence (an)n≥0, defined by a0:= 0, a1:= 1, and a2n:= an, a2n+1:= an + an+1 for any integer n > 0, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of A(α) for every algebraic α with 0 < |α| < 1, and this result was generalized in [6] to the effect that, for the same α's, all numbers A(α),A′(α),A″ (α), are algebraically independent. At about the same time, Bacher [4] studied the twisted version (bn) of Stern's sequence, defined by b0:= 0, b1:= 1, and b2n:= -bn, b2n+1:= -(bn + bn+1) for any n > 0. The aim of our paper is to show the analogs on the generating function B(z) of (bn) of the above-mentioned arithmetical results on A(z), to prove the algebraic independence of (z),B(z) over the field C(z), to use this fact to conclude that, for any complex α with 0 < |α| < 1, the transcendence degree of the field Q(α,A(α),B(α)) over Q is at least 2, and to provide rather good upper bounds for the irrationality exponent of A(r/s) and B(r/s) for integers r, s with 0 < |r|