Transcendence of reciprocal sums of binary recurrences
by
Takeshi Kurosawa
Keio University
Coauthors: Tomoaki Kanoko, Iekata Shiokawa

Let {R_n}_{n ≥ 0} be a binary linear recurrence defined by R_n+2=AR_n+1+BR_n (n ≥ 0), where A, B, R₀, R₁ are integers and D = A²+4B > 0. We give necessary and sufficient conditions on the transcendence of the numbers

å k ≥ 0  ' ak Rrk+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 removed the condition assumed in the preceding work that A ≠ 0 and D is not a perfect square.

Date received: January 29, 2008