On the Sum of Reciprocals of Numbers Satisfying a Recurrence Relation of Order s

We discuss the partial infinite sum ∑∞ k=n u −s k for some positive integer n, where uk satisfies a recurrence relation of order s, un = aun−1 + un−2 + · · ·+ un−s (n ≥ s), with initial values u0 ≥ 0, uk ∈ N (0 ≤ k ≤ s− 1), where a and s(≥ 2) are positive integers. If a = 1, s = 2, and u0 = 0, u1 = 1, then uk = Fk is the k-th Fibonacci number. Our results include some extensions of Ohtsuka and Nakamura. We also consider continued fraction expansions that include such infinite sums.