%0 Journal Article
%T Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials
%A Rafael G. Campos
%A Marisol L. Calder¨®n
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/873078
%X We find approximate expressions and for the real and imaginary parts of the th zero of the Bessel polynomial . 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 , and is obtained. It is shown that the resulting complex number is -convergent to for fixed . 1. Introduction The polynomial solutions of the differential equation were studied systematically in [1] by the first time. They are named (generalized) Bessel polynomials and are given explicitly by as it can be shown in [2]. Here, is the Pochhammer symbol and . Many properties as well as applications are associated to this equation; the traveling waves in the radial direction which are solutions of the wave equation in spherical coordinates can be written in terms of the polynomial solutions of (1.1). Also, this equation has application in network and filter design, isotropic turbulence fields, and more (see the monograph [2] or [3¨C14] and references therein for some other results). Among these, several results about the important problem concerning the location of its zeros have been obtained [8¨C11] and in [12], explicit expressions for sum rules and for the homogeneous product sum symmetric functions of zeros of these polynomials are given. On the other hand, the electrostatic interpretation of these zeros as the equilibrium configuration in the complex plane with a logarithmic electric potential and a dipole at the origin has been given in [13], and in [14] it is shown that this equilibrium configuration is not stable. Thus, these cases show that it is desirable to acquire new analytical knowledge about the location of the zeros of the Bessel polynomials. In this paper we give approximate explicit formulas for both the real and imaginary parts of the th zero of and show that the approximation order of these new formulas to the exact zeros of the Bessel polynomials is for fixed . The approach followed in this paper is simple and based on three items. The first is the electrostatic interpretation of the zeros of polynomials satisfying second-order differential equations [15¨C17], the second is a simple curve fitting of numerical data, and the third is the known fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials
%U http://www.hindawi.com/journals/ijmms/2012/873078/