Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers

摘要

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 ζ_F(2s)?=Σ_(n=1)~∞F_n~(?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 ζ_L(2s)?=Σ_(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 {sum from n=1 to ∞ of 1/F_n~(2s_1), sum from n=1 to ∞ of (-1)~(n+1)/F_n~(2s_2), sum from n=1 to ∞ of 1/L_n~(2s_3), sum from n=1 to ∞ of (-1)~(n+1)/L_n~(2s_4): s1,s2,s3,s4 ∈ N},}and decide on their algebraic independence over . Actually this is a special case of a more general theorem for reciprocal sums of binary recurrent sequences.