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,t given by ${\{0\} = 0, \{1\} = 1}$, and ${\{n \} = s\{n - 1 \} +t\{n - 2 \}}$ for ${{n \geq 2}}$ . The latter are defined by ${\left\{\begin{array}{ll} n\\ k \end{array}\right\} = \{ n \}! / (\{ k \}!\{ n - k \}!)}$ where ${{\{n \}! = \{1 \}\{2 \}\cdots\{n \}}}$. These quotients are also polynomials in s, 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.