Generalized Fibonacci Polynomials and Fibonomial Coefficients

By T. Amdeberhan, X. Chen, Victor H. Moll, Bruce E. Sagan. 2013

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The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials $${\{n\}}$${n}in variables s,t given by $${\{0\} = 0, \{1\} = 1}$${0}=0,{1}=1, and $${\{n \} = s\{n - 1 \} +t\{n - 2 \}}$${n}=s{n-1}+t{n-2} for $${{n \geq 2}}$$n≥2 . The latter are defined by $${\left\{\begin{array}{ll} n\\ k \end{array}\right\} = \{ n \}! / (\{ k \}!\{ n - k \}!)}$$nk={n}!/({k}!{n-k}!) where $${{\{n \}! = \{1 \}\{2 \}\cdots\{n \}}}$${n}!={1}{2}⋯{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 $${\{n\}}$${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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Statistical and Nonlinear Physics

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