Series of reciprocal products with factors from linear recurrence sequences

Abstract

In this article we study the values of power series P z n = Q j i¼0 W ðnþimÞk at certain points of their domain of meromorphy from the arithmetical point of view. ðW n Þ is a sequence of non-zero integers satisfying a recurrence W nþ1 ¼ pW n þ qW nÀ1 with non-zero integers p, q such that the discriminant D ¼ p 2 þ 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 Qð ffiffiffi ffi D p Þ or even in Q, but we also include some transcendence problems.