Negativity Subscripted Fibonacci And Lucas Numbers And Their Complex Factorizations

In this paper, we …nd families of (0; 1; 1) tridiagonal matrices whose determinants and permanents equal to the negatively subscripted Fibonacci and Lucas numbers. Also we give complex factorizations of these numbers by the …rst and second kinds of Chebyshev polynomials. 1. Introduction The well-known Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 2 Fn+1 = Fn + Fn 1 (1.1) where F1 = F2 = 1: The Lucas Sequence, fLng ; is de…ned by the recurrence relation, for n 2 Ln+1 = Ln + Ln 1 (1.2) where L1 = 1; L2 = 3: Rules (1.1) and (1.2) can be used to extend the sequence backward, respectively, thus F 1 = F1 F0; F 2 = F0 F 1 L 1 = L1 L0; L 2 = L0 L 1; : : : ; and so on. Clearly F n = F n+2 F n+1 = ( 1) Fn; (1.3) L n = L n+2 L n+1 = ( 1) Ln: (1.4) In [9] and [5], the authors give complex factorizations of the Fibonacci numbers by considering the roots of Fibonacci polynomials as follows