Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

Abstract

We find approximate expressionsx(k, n) andỹ(k, n) for the real and imaginary parts of the kth zero z k = x k + iy k of the Bessel polynomial y n (x). To obtain these closed-form formulas we use the fact that the points of welldefined 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 numberx(k, n) + iỹ(k, n) is O(1/n 2 )-convergent to z k for fixed k.

Infinite Series Based on Bessel Zeros

“…Presented in this paper, the theory may contribute to the development of many practical issues found in applied science fields, such as (a) the electronics theory of passive n-parts (chapter 6 of [27]); (b) the analysis of time-domain responses of electrical networks and the synthesis of these circuits (amplifiers, etc.) with specified time domain characteristics [28][29][30]; (c) in communication circuits with an emitter current, the output voltages and average transconductance ratio of the modified Bessel function play the major role [31]; (d) analytical solutions for the transient and steady-state responses of linear time-invariant networks [32]; (e) finding the mean charge in graphene nanodots [33]; (f) dynamic analogy in the framework of a linear theory of viscoelasticity [34]; (g) electrical analogs of mechanical models [35]; (h) in mathematical fields, approximate expressions for Bessel functions zero, using a similar technique based on nonlinear equations for the zeros [36,37], the zeros of the equations arising in spectral problems [13,38], the identities between similar summation formulas [12], and important bounds [39,40] are found. The Calogero type series (some of which can be found in the upper part of Table A1) have been recently very useful in finding an inverse Laplace transform of a complicated frequency domain function [41].…”

Computation of the reverse generalized Bessel polynomials and their zeros

“…A first test of the expansion (3) is given in Table 1, where the values of f exact and g exact have been computed with Maple using the relation given in Equation (7) and a large number of digits. For computing the expansion, three coefficients (A 0 (z), A 1 (z), A 2 (z)) have been used.…”

“…On the other hand, several papers can be found in the literature on the approximation and study of the distribution of the zeros of generalized Bessel and reverse generalized Bessel polynomials; see, for example, References 6,7. In this article we discuss the numerical performance of an algorithm to compute the zeros of reverse generalized Bessel polynomials 𝜃 n (z; a). For this, we use the method given in Reference 8, which is based on a qualitative analysis of the approximate Liouville-Green Stokes lines and anti-Stokes lines of the differential equation satisfied by the polynomials, combined with the application of a high-order fixed-point method and carefully selected step functions.…”