The Irreducible Factorization of Fibonacci Polynomials

where a, b and (yn)n>o take values in some specified ring and a and b are fixed elements which do not depend on the Integer index n. A solution (yn)„^0 ^ completely specified once the values of yQ and yx are given. It is customary to denote by Fn(a, b) the solution corresponding to the choice y0 = 0, yt = l, and by Ln(a, b) the solution corresponding to the choice y0 = 2, yx = a. Loosely speaking, in cases of interest the general solution of (1) can be expressed as a linear combination of these two linearly independent solutions. Choosing a = x to be an indeterminate and b to be some fixed integer, each of these solutions defines an infinite sequence of polynomials over Z. More specifically, four distinct polynomial sequences of this type will be considered in the present paper, namely, Un(x) = Fn(x, 1), V„(x) = Ln(x,T), Cn(x) = Fn(x, -1), and Dn(x) = Ln(x, -1). The polynomials Un(x) are known as the Fibonacci polynomials (Un(l) are the Fibonacci numbers), Vn(x) are termed the Lucas polynomials (Vn(T) are the Lucas numbers), while C„(x) and Dn(x) are related to the Chebyshev polynomials. Hereinafter, Fibonacci polynomials" will be used as a collective name for Un(x), Vn{x), Q(x), and £>„(*). The main result of the paper is the prime factorization of the Fibonacci polynomials over the field of rational numbers Q. Webb and Parberry [5] have observed that while Un(x) enjoy all the well-known divisibility properties of the Fibonacci numbers, they possess a general property which the U„(l) lack, namely, that Up(x) is irreducible over Q iffp Is a prime. We recall that the prime factorization of Tn(x) = J^'QX over Q Is well known [4] and, in particular, Tp(x) Is Irreducible over Q iffp is a prime. The irreducible factors are the cyclotomic polynomials <&m(x), which are given by