On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

The aim of this paper is to give new results about factorizations of the Fibonacci numbers F _n and the Lucas numbers L _n . These numbers are defined by the second order recurrence relation a _n+2 = a _n+1+a _n with the initial terms F ₀ = 0, F ₁ = 1 and L ₀ = 2, L ₁ = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].