On The Generalized Fibonacci And Pell Sequences By Hessenberg Matrices

In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants. 1. Introduction The Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 1 Fn+1 = Fn + Fn 1 (1.1) where F0 = 0; F1 = 1: The Pell Sequence, fPng ; is de…ned by the recurrence relation, for n 1 Pn+1 = 2Pn + Pn 1 (1.2) where P0 = 0; P1 = 1: The well-known Fibonacci and Pell numbers can be generalized as follow: Let A be nonzero, relatively prime integers such that D = A +4 6= 0: De…ne sequence fung by, for all n 2 (see [17]), un = Aun 1 + un 2 (1.3) where u0 = 0; u1 = 1: If A = 1; then un = Fn (the nth Fibonacci number). If A = 2; then un = Pn ( the nth Pell number). An alternative is to let the roots of the equation t At 1 = 0 be, for n 0 un = n n : The sequence fung have studied by several authors (see [6], [1]). The following identities can be found in [6], [1]: