Release Time:2019-03-09  Hits:

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: 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.