Algebraic independence of the generating functions of the Stern polynomials and their twisted analogues

Abstract

Dilcher and Stolarsky introduced and studied a polynomial analogue of Stern’s diatomic sequence. Similarly, we define here a polynomial analogue of Bacher’s twisted version of the Stern sequence. Our main aim is to consider the two-dimensional generating functions of both these polynomial sequences. More precisely, since they satisfy certain Mahler-type functional equations, we succeed in characterizing the algebraic independence over \({\mathbb{Q}}\) of the values of these two functions at algebraic points \({(\xi,\alpha)}\) with \({0 < |\xi|,|\alpha| < 1}\).