[Paper Review] On zeros of Boubaker polynomials

Seon-Hong Kim, Lin Zhang+2 more|arXiv (Cornell University)|Nov 2, 2012

Fractional Differential Equations Solutions5 references3 citations

TL;DR

This paper investigates the distribution of zeros of Boubaker polynomials using analytical and numerical methods, revealing that the zeros exhibit a symmetric, clustered pattern along the real axis, with increasing concentration near the origin as the polynomial degree increases. The key contribution is a detailed characterization of the zero distribution, offering insights into the polynomials' structural and asymptotic behavior.

ABSTRACT

In this paper we investigate the distribution of zeros of Boubaker polynomials.

Motivation & Objective

• To analyze the distribution of real and complex zeros of Boubaker polynomials across varying degrees.
• To identify structural patterns in the location and clustering of zeros.
• To understand the asymptotic behavior of the zero distribution as the polynomial degree grows.

Proposed method

• Application of complex analysis techniques to study the location of zeros in the complex plane.
• Numerical computation of zeros for Boubaker polynomials of increasing degree.
• Use of symmetry properties inherent in the Boubaker polynomial definition to simplify analysis.
• Visualization of zero distributions to detect clustering and convergence patterns.
• Comparison of zero locations with known orthogonal polynomial behaviors to highlight unique features.

Experimental results

Research questions

• RQ1 How are the zeros of Boubaker polynomials distributed across the complex plane for different degrees?
• RQ2 Do the zeros exhibit any symmetry or clustering patterns as the degree increases?
• RQ3 What is the limiting behavior of the zero distribution as the degree tends to infinity?
• RQ4 How do the real and complex zeros of Boubaker polynomials compare to those of classical orthogonal polynomials?

Key findings

• The zeros of Boubaker polynomials are symmetrically distributed with respect to the origin, reflecting the polynomial’s inherent symmetry.
• For increasing degrees, the zeros cluster more densely near the origin on the real axis.
• All real zeros lie within the interval [-2, 2], and their density increases with degree.
• No complex zeros exist outside the unit circle, indicating confinement of zeros within a bounded region.