# Reciprocal Sums of Elements Satisfying Second Order Linear Recurrences

## Authors

• Kantaphon Kuhapatanakul
• Vichian Laohakosol

## Keywords:

reciprocal sum, linear recurrence, Fibonacci numbers

## Abstract

Let $\{U_{n}\}^{\infty}_{n=0}$ and $\{W_{n}\}^{\infty}_{n=0}$ be two sequences defined by $U_{0}=0$, $U_{1}=1$, $U_{n+2}=pU_{n+1}+qU_{n}$ and $W_{n+2}=pW_{n+1}+qW_{n}$ ($W_{0}$, $W_{1}$ arbitrary) with $p,q&space;\in&space;\mathbb{R}$; $p^{2}+4q$ > $0$. The aim of this paper is to prove

\begin{align*}&space;\sum_{n=1}^{N}\dfrac{(-q)^{at-1}U_{at^{n}(t-1)}}{W_{at^{n}}W_{at^{n+1}}}&space;&=&space;\sum_{n=1}^{_{at}N+1_{-at}}\dfrac{(-q)^{at-1}p}{W_{at+2(n-1)}W_{at+2n}}&space;\\&space;&=\dfrac{1}{W_{0}W_{2}-W_{1}^{2}}\left(\dfrac{W_{at-1}}{W_{at}}-\dfrac{W_{at^{N+1}-1}}{W_{at^{N+1}}}\right&space;),&space;\end{align*}

where $a,t&space;\in&space;\mathbb{N}$ and $t&space;\geq&space;2$. This identity generalizes a number of known identities such as $\sum_{n=0}^{\infty}\frac{1}{F_{2^{n}}}=\frac{7-\sqrt{5}}{2}$, where $\{F_{n}\}$ is Fibonacci sequence.