Series of reciprocal products with factors from linear recurrence sequences

Abstract

In this article we study the values of power series $\sum z^n/{\prod }_{i=0}^j{W}_{(n+im)k}$ at certain points of their domain of meromorphy from the arithmetical point of view. $(W_n)$ is a sequence of non-zero integers satisfying a recurrence $W_{n+1}=p{W}_n+q{W}_{n-1}$ with non-zero integers $p,q$ such that the discriminant $\Delta =p^2+4q$ is positive but not a square. The main interest is to characterize the situations, where these values lie in the real quadratic number field $Q(\surd \Delta )$ or even in $Q$, but we also include some transcendence problems.