Algebraic relations for reciprocal sums of even terms in Fibonacci numbers

Abstract

In this paper, we discuss the algebraic independence and algebraic relations, first, for reciprocal sums of even terms in Fibonacci numbers $\sum\limits_{n = 1}^\infty {F_{2n}^{ - 2s}}$, and second, for sums of evenly even and unevenly even types $\sum\limits_{n = 1}^\infty {F_{4n}^{ - 2s}}, \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}}$. The numbers $\sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}}, \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 4}}$, and are shown to be algebraically independent, and each sum $\sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}} \left( {s \geqslant 4} \right)$ 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 $\sum\limits_{n = 1}^\infty {L_{2n}^{ - p}}, \sum\limits_{n = 1}^\infty {L_{4n}^{ - p}}$, and $\sum\limits_{n = 1}^\infty {L_{4n - 2}^{ - p}}$.