Transcendence of Hecke-Mahler Series

TL;DR

This paper proves the transcendence of a class of Hecke-Mahler series involving polynomial functions, irrational and algebraic parameters, advancing understanding in transcendence theory.

Contribution

It establishes the transcendence of Hecke-Mahler series with polynomial coefficients and irrational, algebraic parameters, a novel result in number theory.

Findings

Proves transcendence of specific Hecke-Mahler series

Extends transcendence results to series with polynomial functions

Provides new techniques for transcendence proofs in series

Abstract

We prove transcendence of the Hecke-Mahler series $\sum_{n=0}^\infty f(\lfloor n\theta+\alpha \rfloor) \beta^{-n}$, where $f(x) \in \mathbb{Z}[x]$ is a non-constant polynomial $\alpha$ is a real number, $\theta$ is an irrational real number, and $\beta$ is an algebraic number such that $|\beta|>1$.