Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers

Let F(n) and L(n) 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 zeta(F)(2s) = Sigma(infinity)(n=1) F(n)(-2s) are transcendental for any s is an element of N using Nesterenko's theorem on Ramanujan functions P(q), Q(q), and R(q). They obtained similar results for the Lucas zeta function zeta(L) (2s) = Sigma(infinity)(n=1) L(n)(-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 {Sigma(infinity)(n=1) 1/F(n)(2s1), Sigma(infinity)(n=1) (-1)(n+1)/F(n)(2s2), Sigma(infinity)(n=1) 1/L(n)(2s3), Sigma(infinity)(n=1) (-1)(n+1)/F(n)(2s4) : s(1), s(2), s(3), s(4) is an element of N} 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.