A linear independence criterion for certain infinite series with polynomial orders

TL;DR

This paper establishes a criterion for linear independence over algebraic number fields for certain infinite series involving polynomial orders and algebraic integers, extending understanding of their algebraic independence.

Contribution

It introduces a new linear independence criterion for infinite series with polynomial orders and algebraic integer coefficients over Q(q), especially when coefficients are polynomials.

Findings

Provides a criterion for linear independence over Q(q).

Applies to series with polynomial order functions.

Extends previous results on algebraic independence.

Abstract

Let $q$ be a Pisot or Salem number. Let $f_j(x)$ $(j=1,2,\dots)$ be integer-valued polynomials of degree $\ge2$ with positive leading coefficients, and let $\{a_j (n)\}_{n\ge1}$ $(j=1,2,\dots)$ be sequences of algebraic integers in the field $\mathbb{Q}(q)$ with suitable growth conditions. In this paper, we investigate linear independence over $\mathbb{Q}(q)$ of the numbers \begin{equation*} 1,\qquad \sum_{n=1}^{\infty} \frac{a_j (n)}{q^{f_j (n)}} \quad (j=1,2,\dots). \end{equation*} In particular, when $a_j(n)$ $(j=1,2,\dots)$ are polynomials of $n$, we give a linear independence criterion for the above numbers.