The sums of the reciprocals of Fibonacci polynomials and Lucas polynomials

Zhang, Wu Zhengang, Zhang Wenpeng

Description

In this article, we consider infinite sums derived from the reciprocals of the Fibonacci polynomials and Lucas polynomials, and infinite sums derived from the reciprocals of the square of the Fibonacci polynomials and Lucas polynomials. Then applying the floor function to these sums, we obtain several new equalities involving the Fibonacci polynomials and Lucas polynomials. Mathematics Subject Classification (2010): Primary, 11B39.

Sujets

• Fibonacci polynomials
• Lucas number
• Inequality
• Floor and ceiling functions

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 13
• Langue: English

Extrait

Wu and ZhangJournal of Inequalities and Applications2012,2012:134 http://www.journalofinequalitiesandapplications.com/content/2012/1/134

R E S E A R C HOpen Access The sums of the reciprocals of Fibonacci polynomials and Lucas polynomials * Zhengang Wuand Wenpeng Zhang

* Correspondence: sky.wzgfff@163. com Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China

Abstract In this article, we consider infinite sums derived from the reciprocals of the Fibonacci polynomials and Lucas polynomials, and infinite sums derived from the reciprocals of the square of the Fibonacci polynomials and Lucas polynomials. Then applying the floor function to these sums, we obtain several new equalities involving the Fibonacci polynomials and Lucas polynomials. Mathematics Subject Classification (2010):Primary, 11B39. Keywords:Fibonacci polynomials, Lucas polynomials, inequality, floor function

1. Introduction For any variable quantityx, the Fibonacci polynomialsF n (x) and Lucas polynomialsL n (x) are defined byF n+2 (x) =xF n+1 (x) +F n (x),n≥0 with the initial valuesF 0 (x) = 0 and F 1 (x) = 1;L n+2 (x) =xL n+1 (x) +L n (x),n≥0 with the initial valuesL 0 (x) = 2 andL 1 (x) = x. Forx= 1 we obtain the usual Fibonacci numbers and Lucas numbers. Let    √ 1 1 2 2 α= x+ x+ 4 and β= x− x+ 4 , then from the properties of the sec 2 2 ondorder linear recurrence sequences we have n n α −β n n F n ( x ) = andL n ( x ) =α+β. 2 x+ 4 Various authors studied the properties of Fibonacci polynomials and Lucas polyno mials, and obtained many interesting results, see [13]. Recently, several authors studied the infinite sums derived from the reciprocals of the Fibonacci numbers and Pell num bers, and obtained some important results. For example, Ohtsuka and Nakamura [4] stu died the properties of the Fibonacci numbers, and proved the following conclusions:   ∞  −1    1 F n−2 , ifnis even andn≥2;   = F k F n−2 −1, ifnis odd andn≥1. k=n

  ∞  −1    1 F n−1 F n −1, ifnis even andn≥2;   = 2 F n−1 F n , ifnis odd andn≥1. F k k=n

© 2012 Zhengang and Wenpeng; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.