Generalized Fibonacci Polynomials and Fibonomial Coefficients

Abstract

The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s, t given by {0} = 0, {1} = 1, and {n} = s{n−1}+t{n−2} for n ≥ 2. The latter are defined byThese 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}, 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.

A Combinatorial Proof of a Result on Generalized Lucas Polynomials

“…The main aim of this paper is to give a combinatorial proof of the following result, inspired by [1].…”

“…For some historical remarks and relations of these polynomials we refer the reader to [1], [2] and [3].…”

“…We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation xny " sxn´1y`txn´2y for n ≥ 2.…”

ON SECOND-ORDER LINEAR RECURRENT FUNCTIONS WITH PERIOD k AND PROOFS TO TWO CONJECTURES OF SROYSANG

“…In [11], the author presented a formula for solving the missing terms of {W n } given its first term and last term. Another generalization of Fibonacci numbers is the so-called Fibonacci polynomials (see [1] and [3] and the references therein). Recently, Fibonacci numbers were involved in the study of difference and differential equations.…”

On The Lagrange Interpolation of Fibonacci Sequence

“…In mathematics, Fibonacci occurs in many fields for instance in linear algebra, combinatorics, and discrete mathematics. There are some developments about Fibonacci sequence, for example Fibonomial (Fibonacci and Binomial) [2], [3] and Fibonacci Polynomial [4], [3].…”