Generalized Fibonacci polynomials and Fibonomial coefficients

TL;DR

This paper explores generalized Fibonacci polynomials and Fibonomial coefficients, establishing their fundamental properties, identities, and applications, while proposing conjectures and open problems for further research.

Contribution

It introduces a unified framework for Fibonacci polynomials and Fibonomial coefficients, deriving new identities and generalizations, including a binomial theorem analogue and Catalan number extension.

Findings

Derived a general recursion for Fibonacci polynomials

Established an analogue of the binomial theorem

Proved a new Euler-Cassini identity and explored valuations

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.