A linear independence criterion for certain infinite series with polynomial orders

Let q be a Pisot or Salem number. Let \(f_j(x) \quad (j=1,2,\dots)\) be integer-valued polynomials of degree \(\ge2\) with positive leading coefficients, and let \(\{a_j (n)\}_{n\ge1} \quad (j=1,2,\dots)\) be sequences of algebraic integers in the field \(Q(q)\) with suitable growth conditions. In this paper, we investigate linear independence over \(Q(q)\) of the numbers

$$1,\quad \sum_{n=1}^{\infty} \frac{a_j (n)}{q^{f_j (n)}} \quad (j=1,2,\dots).$$

In particular, when \(a_j(n) \quad (j=1,2,\dots)\) are polynomials of n, we give a linear independence criterion for the above numbers.