Reciprocal Sums of Elements Satisfying Second Order Linear Recurrences

Kantaphon Kuhapatanakul; Vichian Laohakosol

Authors

Kantaphon Kuhapatanakul
Vichian Laohakosol

Keywords:

reciprocal sum, linear recurrence, Fibonacci numbers

Abstract

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

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

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

Reciprocal Sums of Elements Satisfying Second Order Linear Recurrences

Authors

Keywords:

Abstract

Downloads

Published

Issue

Section

Make a Submission