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}.

Location: http://mech.math.msu.su/~fpm/eng/k10/k105/k10513h.htm

Last modified: May 30, 2011