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arXiv · 1306.6511

Generalized Fibonacci polynomials and Fibonomial coefficients

Tewodros Amdeberhan·Xi Chen·Victor H. Moll·Bruce E. Sagan
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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} = s{n-1}+t{n-2} for n ge 2. The latter are defined by {n choose k} = {n}!/({k}!{n-k}!) where {n}! = {1}{2}...{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.

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Tewodros Amdeberhan, Xi Chen, Victor H. Moll, Bruce E. Sagan. 2013-06-27. Generalized Fibonacci polynomials and Fibonomial coefficients. https://arxiv.org/abs/1306.6511

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