Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

摘要

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers Sigma(infinity)(n=1) F-2n-1(-1), Sigma(infinity)(n=1) F-2n-1(-2) (n=1), Sigma(infinity)(n=1) F(2n-1)(-3)and write each Sigma(infinity)(n=1) F-2n-1(-s) (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.