On the sum of reciprocals of numbers satisfying a recurrence relation of order \(s\)

Komatsu, Takao ; Laohakosol, Vichian

Journal of integer sequences, Tome 13 (2010) no. 5

Résumé

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.

Détail

Classification : 11A55, 11B39
Keywords: Fibonacci numbers, recurrence relations of s-th order, partial infinite sum