Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1 ^∞ F _2n-1 ^-1 , ∑ _n=1 ^∞ F _2n-1 ^-2 , ∑ _n=1 ^∞ F _2n-1 ^-3 and write each ∑ _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.