A linear independence criterion for certain infinite series with polynomial orders

Let $q$ be a Pisot or Salem number. Let $f_j(x)$ $(j=1,2,\dots )$ be integer-valued polynomials of degree $\ge 2$ with positive leading coefficients, and let $\{a_j(n){\}}_{n\ge 1}$ $(j=1,2,\dots )$ 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.