Generalized Compositions and Weighted Fibonacci Numbers

TL;DR

This paper explores generalized compositions of natural numbers, linking them to weighted polynomial coefficients and r-generalized Fibonacci numbers, thereby extending classical Fibonacci-binomial relationships.

Contribution

It introduces a new connection between generalized compositions, weighted polynomial coefficients, and r-generalized Fibonacci numbers, generalizing classical Fibonacci identities.

Findings

Number of generalized compositions equals weighted polynomial coefficients.

Total generalized compositions relate to weighted r-generalized Fibonacci numbers.

Derived a generalized Fibonacci-binomial coefficient relationship.

Abstract

In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted $r$-generalized Fibonacci number. A relationship between these two numbers will be derived. We shall thus obtain a generalization of the well-known formula connecting Fibonacci numbers with the binomial coefficients.