Oresme Polynomials and Their Derivatives

TL;DR

This paper introduces Oresme polynomials as a generalization of Oresme numbers, explores their identities using matrix methods, and studies their derivatives and relations, extending the classical sequence with new algebraic properties.

Contribution

The paper presents a novel generalization of Oresme numbers through Oresme polynomials, deriving identities and derivative relations using matrix techniques.

Findings

Derived bilinear index-reduction formulas for Oresme polynomials

Introduced natural extensions called $k$-Oresme polynomials

Established convolution relations for derivatives of these polynomials

Abstract

We study the problem of generalization of Oresme numbers with a new sequence of numbers called Oresme polynomials. Moreover, by using the matrix methods for Oresme polynomials, we obtain the identities including the general bilinear index-reduction formula of these numbers. Further, Oresme polynomials that are natural extensions of the $k$-Oresme numbers are introduced and some relations for the derivatives of these polynomials in the form of convolution are proved.