FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 5, PAGES 173-200

Algebraic relations for reciprocal sums of even terms in Fibonacci numbers

C. Elsner
Sh. Shimomura
I. Shiokawa

Abstract

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In this paper, we discuss the algebraic independence and algebraic relations, first, for reciprocal sums of even terms in Fibonacci numbers $$å_n=1^¥ F_2n^-2s, and second, for sums of evenly even and unevenly even types $$å_n=1^¥ F^-2s_4n, $$å_n=1^¥ F^-2s_4n-2. The numbers $$å_n=1^¥ F_4n-2^-2, $$å_n=1^¥ F_4n-2^-4, and $$å_n=1^¥ F_4n-2^-6 are shown to be algebraically independent, and each sum $$å_n=1^¥ F^-2s_4n-2 ($s$³ 4) is written as an explicit rational function of these three numbers over $$Q. Similar results are obtained for various series of even type, including the reciprocal sums of Lucas numbers $$å_n=1^¥ L_2n^-p, $$å_n=1^¥ L^-p_4n, and $$å_n=1^¥ L^-p_4n-2.

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