Series of reciprocal products with factors from linear recurrence sequences

Abstract

.ions { line-height: 1.8; } .ions sub { margin-left: -.5ex; } .ions sup { vertical-align: 1.2ex; } In this article we study the values of power series ∑ zn/∏ji = 0 W(n + im)k at certain points of their domain of meromorphy from the arithmetical point of view. (Wn) is a sequence of non-zero integers satisfying a recurrence Wn + 1 = pWn + qWn − 1 with non-zero integers p, q such that the discriminant Δ = p2 + 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(√Δ) or even in ℚ, but we also include some transcendence problems.