Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers

Let $Fn$ and $Ln$ denote the Fibonacci and Lucas numbers, respectively. D. Duverney, Ke. Nishioka, Ku. Nishioka and I. Shiokawa proved that the values of the Fibonacci zeta function $ζF(2s) = Σn = 1∞Fn−2s$ are transcendental for any $s∈N$ using Nesterenko’s theorem on Ramanujan functions $P(q),$ $Q(q),$ and $R(q).$ They obtained similar results for the Lucas zeta function $ζL(2s) = Σn = 1∞Ln−2s$ and some related series. Later, C. Elsner, S. Shimomura and I. Shiokawa found conditions for the algebraic independence of these series. In my PhD thesis I generalized their approach and treated the following problem: We investigate all subsets of and decide on their algebraic independence over ℚ. Actually this is a special case of a more general theorem for reciprocal sums of binary recurrent sequences.