Abstract:
In this paper, we discuss the algebraic independence and algebraic relations, first, for reciprocal sums of even terms in Fibonacci numbers $\sum^\infty_{n=1} F_{2n}^{-2s}$, and second, for sums of evenly even and unevenly even types $\sum^\infty_{n=1}F^{-2s}_{4n}$, $\sum^\infty_{n=1}F^{-2s}_{4n-2}$. The numbers $\sum^\infty_{n=1}F_{4n-2}^{-2}$, $\sum^\infty_{n=1}F_{4n-2}^{-4}$, and $\sum^\infty_{n=1}F_{4n-2}^{-6}$ are shown to be algebraically independent, and each sum $\sum^\infty_{n=1}F^{-2s}_{4n-2}$ ($s\ge4$) is written as an explicit rational function of these three numbers over $\mathbb Q$. Similar results are obtained for various series of even type, including the reciprocal sums of Lucas numbers $\sum^\infty_{n=1}L_{2n}^{-p}$, $\sum^\infty_{n=1}L^{-p}_{4n}$, and $\sum^\infty_{n=1}L^{-p}_{4n-2}$.
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