SUMS RELATED TO THE FIBONACCI SEQUENCE

We investigate sums associated with the Fibonacci sequence F-n and the golden ratio phi. In particular, we study the sums G(k) = Sigma(infinity)(n=1) n(k)/F-n and H(k) = root 5. Li-k(1/phi) = Sigma(infinity)(n=1) n(k)root 5/phi(n). These sums generalize the reciprocal Fibonacci constant psi = G(0). We prove the asymptotic equivalence G(k) similar to H(k), and moreover, G(k)/H(k) = 1 + 1/5(k + 1) + O((log phi/pi)(k + 1)) as k -> infinity. We express G(k) - H(k) as an alternating series, allowing us to compute values of these sums to high precision, and to prove that G(k) > H(k) if and only if k >= 2. We also generalize the results to their Lucas sequence analogues. As a tool, we establish a widely applicable explicit bound for polylogarithms of negative integer order. We find explicit bounds for the integer sequences {A(k)}(k=1)(infinity) and {B-k}(k=1)(infinity) defined by H(k)/root 5 =Li-k (1/phi)= A(k)+B-k phi. We also prove several results concerning the multiplicative structure of A(k) and B-k. We show that {A(k) (mod m)} and {B-k (mod m)} are periodic for every natural number m, and that the period is a divisor of lambda(m), where lambda denotes the Carmichael function.