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 , where Ek(z), Fk(z) are polynomials, α 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 , where Rn is a binary linear recurrence and {ak} is a sequence of algebraic numbers.