Transcendence of certain series involving binary linear recurrences

Duverney and Nishioka [D. Duverney, Ku. Nishioka, An inductive method for proving the transcendence of certain series, Acta Arith. 110 (4) (2003) 305-330] studied the transcendence of Sigma(k >= 0) Ek(alpha(rk))/Fk(alpha(rk)), where E-k (z), F-k, (z) are polynomials, alpha is an algebraic number, and r is an integer greater than 1, using an inductive method. We extend their inductive method to the case of several variables. This enables us to prove the transcendence of Sigma(k >= 0)' alpha(k)/R-cr(+d)k where& is a binary linear recurrence and {a(k)} is a sequence of algebraic numbers. (c) 2006 Elsevier Inc. All rights reserved.