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Indexed by:Journal Papers

Date of Publication:2014-12-01

Journal:ANNALS OF COMBINATORICS

Included Journals:SCIE

Volume:18

Issue:4

Page Number:541-562

ISSN No.:0218-0006

Key Words:binomial theorem; Catalan number; Dodgson condensation; Euler-Cassini identity; Fibonacci number; Fibonomial coefficient; Lucas number q-analogue; valuation

Abstract:The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials in variables s,t given by , and for . The latter are defined by where . 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 , 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.