Fibonacci numbers and trigonometric identities

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Fibonacci numbers and trigonometric identities N. Garnier Universite Lille 1, 59 655 Villeneuve d'Ascq, France O. Ramare Laboratoire Paul Painleve, Universite Lille 1, 59 655 Villeneuve d'Ascq, France (Submitted April 2006)? Abstract Webb & Parberry proved in 1969 a startling trigonometric iden- tity involving Fibonacci numbers. This identity has remained isolated up to now, despite the amount of work on related polynomials. We provide a wide generalization of this identity together with what we believe (and hope!) to be its proper understanding. 1. Introduction Fibonacci numbers verify a wealth of identities, see e.g. [5], [6], [7], [11]. By specifying x and y to 1 in Corollary 10 of [4], we get an intriguing one which states that for n ≥ 1: Fn = [(n?1)/2]∏ k=1 ( 1 + 4 cos2 kpi n ) . (1) Webb & Parberry's paper [14] contains already all the necessary material to write this identity, but they do not state it explicitly. This formula is indeed intriguing: the left hand side satisfies a second order recursion formula while no such recursion arises from the right hand side expression. Indeed, how could we connect cos 2kpin and cos 2kpi n+1? Taking a number theoretic point of view leads to more dismay: Fibonacci numbers are linked with the arithmetic of Q( √ 5) and not

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Publié par

profil-urra-2012

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English

Fibonacci numbers and identities

trigonometric

N. Garnier Universite´Lille1,59655Villeneuved’Ascq,France O.Ramare´ LaboratoirePaulPainlev´e,Universit´eLille1,59655Villeneuved’Ascq,France ∗ (Submitted April 2006)

Abstract Webb & Parberry proved in 1969 a startling trigonometric iden-tity involving Fibonacci numbers. This identity has remained isolated up to now, despite the amount of work on related polynomials. We provide a wide generalization of this identity together with what we believe (and hope!) to be its proper understanding.

Introduction

Fibonacci numbers verify a wealth of identities, see e.g. [5], [6], [7], [11]. By specifyingxandyto 1 in Corollary 10 of [4], we get an intriguing one which states that forn≥1: [(n−1)/2]  Y kπ 2 Fn+ 4 cos= 1 .(1) n k=1 Webb & Parberry’s paper [14] contains already all the necessary material to write this identity, but they do not state it explicitly. This formula is indeed intriguing: the left hand side satisﬁes a second order recursion formula while no such recursion arises from the right hand side expression. Indeed, how 2kπ2kπ could we connect cos and cos ? Taking a number theoretic point of n n+1 view leads to more dismay: Fibonacci numbers are linked with the arithmetic √ ofQ( 5) and not with that ofQ(exp(2iπ/n)). ∗ AMS Classiﬁcation: 11B39

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