Abstract

We discuss the partial infinite sum ∑_{k=n}^{∞} u_{k}^{−s} for some positive integer n, where u_{k} satisfies a recurrence relation of order s, u_{n} = au_{n−1} + u_{n−2} + ⋯ + u_{n−s} (n ≥ s), with initial values u_{0} ≥ 0, u_{k} ∈ ℕ (0 ≤ k ≤ s − 1), where a and s(≥ 2) are positive integers. If a = 1, s = 2, and u_{0} = 0, u_{1} = 1, then u_{k} = F_{k} 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.