, Volume 157, Issue 4, pp 323-334

Transcendence of reciprocal sums of binary recurrences

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Abstract

Let {R n } n≥0 be a binary linear recurrence defined by R n+2 = A R n+1 + B R n (n ≥ 0), where A, B, R 0, R 1 are integers and Δ = A 2 + 4B > 0. We give necessary and sufficient conditions for the transcendence of the numbers $$\sum_{k\geq 0}{}^{\prime}\frac{a_k}{R_{r^k}+b},$$ where r ≥ 2 is an integer, {a k } k ≥ 0 is a linear recurrence of algebraic numbers, and b is an algebraic number. We remove the condition assumed in the preceding work that A ≠ 0 and Δ is not a perfect square.

Communicated by U. Zannier.