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 \(\ge2\) 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.