Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

Abstract

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ =1 ∞ F 2−1 −1 , ∑ =1 ∞ F 2−1 −2 , ∑ =1 ∞ F 2−1 −3 and write each ∑ =1 ∞ F 2−1 − (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.