Generalized Fibonacci polynomials and Fibonomial coefficients
Abstract
The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials n in variables s and t given by 0 = 0, 1 = 1, and n = sn-1+tn-2 for n ge 2. The latter are defined by n choose k = n!/(k!n-k!) where n! = 12...n. These quotients are also polynomials in s and t, and specializations give the ordinary binomial coefficients, the Fibonomial coefficients, and the q-binomial coefficients. We present some of their fundamental properties, including a more general recursion for n, an analogue of the binomial theorem, a new proof of the Euler-Cassini identity in this setting with applications to estimation of tails of series, and valuations when s and t take on integral values. We also study a corresponding analogue of the Catalan numbers. Conjectures and open problems are scattered throughout the paper.