Stern polynomials and algebraic independence

TL;DR

This paper proves the algebraic independence of certain special functions related to Stern polynomials at algebraic points, using Mahler's method, contributing to the understanding of their transcendental properties.

Contribution

It introduces a novel application of Mahler's method to establish algebraic independence of functions derived from Stern polynomials.

Findings

Proves algebraic independence of $H_k(z)$ and $H_k(z^{t^k})$ at algebraic points.

Extends Mahler's method to a new class of functions related to Stern polynomials.

Provides insights into the transcendental nature of values associated with Stern polynomials.

Abstract

Let $t\geq2$ and $k\geq1$ be integers. Let $H_{k}(z)$ with $\left\vert z\right\vert <1$ be the limit of a certain subsequence of the Stern polynomials introduced by Dilcher and Eriksen. We use Mahler's method to prove the algebraic independence of the values at nonzero algebraic points of the functions $H_{k}(z)$ and $H_{k}(z^{t^{k}})$.