Approximate closed-form formulas for the zeros of the Bessel Polynomials

TL;DR

This paper derives approximate closed-form formulas for the zeros of Bessel polynomials by numerically computing zeros and fitting their real and imaginary parts as functions of degree and zero index, achieving O(1/n^2) convergence.

Contribution

The paper introduces a novel method to approximate Bessel polynomial zeros using numerical computation and curve fitting, providing explicit formulas with proven convergence.

Findings

Formulas accurately approximate zeros for large n.

Achieves O(1/n^2) convergence to actual zeros.

Provides a practical approach for analyzing Bessel polynomial zeros.

Abstract

We find approximate expressions x(k,n) and y(k,n) for the real and imaginary parts of the kth zero z_k=x_k+i y_k of the Bessel polynomial y_n(x). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions of k and n is obtained. It is shown that the resulting complex number x(k,n)+i y(k,n) is O(1/n^2)-convergent to z_k for fixed k