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.
Categories: math.NT