Abstract:

We discuss the partial infinite sum $\sum_{k=n}^{\infty}u_k^{-s}$ for some positive integer $n$, where $u_k$ satisfies a recurrence relation of order $s$, $u_n= a u_{n-1}+u_{n-2}+\cdots+u_{n-s}$ ($n\ge s$), with initial values $u_0\ge 0$, $u_k\in\mathbb{N}$ ( $0\le k\le s-1$), where $a$ and $s(\ge 2)$ 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.