[Paper Review] Oresme Polynomials and Their Derivatives

Gamaliel Cerda-Morales|arXiv (Cornell University)|Apr 2, 2019

Advanced Mathematical Theories and Applications2 references5 citations

TL;DR

This paper introduces Oresme polynomials as a rational function extension of $k$-Oresme numbers, using matrix methods and recurrence relations to derive closed-form Binet-type formulas and convolution identities for their derivatives. The key contribution is a general bilinear index-reduction formula and novel derivative identities involving convolution sums.

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.

Motivation & Objective

• To generalize $k$-Oresme numbers into a sequence of rational functions called Oresme polynomials by replacing the integer parameter $k$ with a real variable $x$.
• To establish matrix representations for Oresme polynomials and derive determinant-based identities, including a generalized index-reduction formula.
• To investigate the derivatives of Oresme polynomials and prove convolution-type identities involving sums of products of polynomials.
• To extend known identities for $k$-Oresme numbers to the polynomial setting using induction and generating function techniques.

Proposed method

• Define Oresme polynomials $O_n(x)$ via a second-order linear recurrence: $O_{n+1}(x) = O_n(x) - \frac{1}{x^2}O_{n-1}(x)$, with initial conditions $O_0(x) = 0$, $O_1(x) = \frac{1}{x}$.
• Derive the Binet formula for $O_n(x)$ using the roots of the characteristic equation $x^2 - x + \frac{1}{x^2} = 0$, yielding $O_n(x) = \frac{1}{\sqrt{x^2 - 4}}\left(\lambda_1^n(x) - \lambda_2^n(x)\right)$.
• Construct a $2 \times 2$ matrix $M_{or}(x) = \begin{bmatrix} 1 & -\frac{1}{x^2} \\ 1 & 0 \end{bmatrix}$ whose powers generate $O_n(x)$ and yield determinant identities.
• Use mathematical induction to prove derivative identities, including $O_n'(x) + \frac{n}{x}O_n(x) = \sum_{j=1}^{n-1} O_j(x)O_{n-j}(x)$.
• Establish a bilinear index-reduction formula via matrix powers and convolution identities, valid for $n \geq 2$ and $x \neq 0, \pm 2$.
• Prove a novel identity relating derivatives and polynomial values: $(n-1)O_n(x) - 2nO_{n+1}(x) = xO_{n+1}'(x) - \frac{1}{x}O_{n-1}'(x)$.

Experimental results

Research questions

• RQ1 How can the $k$-Oresme numbers be generalized to a sequence of rational functions in a real variable $x$?
• RQ2 What matrix representation enables the derivation of closed-form identities for Oresme polynomials?
• RQ3 What convolution identities emerge for the derivatives of Oresme polynomials?
• RQ4 How do the Binet-type formulas for Oresme polynomials relate to Fibonacci numbers when $x = 3$?
• RQ5 What is the asymptotic behavior of the ratio $\frac{O_{n+1}(x)}{O_n(x)}$ as $n \to \infty$?

Key findings

• The Oresme polynomial $O_n(x)$ satisfies the Binet formula $O_n(x) = \frac{1}{\sqrt{x^2 - 4}}\left(\lambda_1^n(x) - \lambda_2^n(x)\right)$ for $x^2 > 4$, where $\lambda_{1,2}(x) = \frac{x \pm \sqrt{x^2 - 4}}{2x}$.
• The matrix $M_{or}^n(x)$ yields the identity $O_{n+1}(x)O_{n-1}(x) - O_n^2(x) = -\frac{1}{x^{2n}}$, generalizing the Cassini-like identity for $k$-Oresme numbers.
• For $x > 2$, the limit $\lim_{n \to \infty} \frac{O_{n+1}(x)}{O_n(x)} = \lambda_1(x)$, the dominant root of the characteristic equation.
• The derivative identity $O_n'(x) + \frac{n}{x}O_n(x) = \sum_{j=1}^{n-1} O_j(x)O_{n-j}(x)$ holds for all $n \geq 2$.
• A bilinear index-reduction formula is derived: $\sum_{j=1}^{n-1} O_j(x)O_{n-j}(x) = \frac{x^2((n-1)x^2 - 2n)O_n(x) - 2nO_{n-2}(x)}{x^3(x^2 - 4)}$ for $n \geq 2$, $x \neq 0, \pm 2$.
• The identity $(n-1)O_n(x) - 2nO_{n+1}(x) = xO_{n+1}'(x) - \frac{1}{x}O_{n-1}'(x)$ is proven by induction for $n \geq 1$.