The Zeros of Special Functions from a Fixed Point Method

Abstract

A scheme for the computation of the zeros of special functions and orthogonal polynomials is developed. We study the structure of the first order difference-differential equations (DDEs) satisfied by two fundamental sets of solutions of second order ODEs $y_n^{\prime\prime}(x)+A_{n}(x)y_n (x)=0$, n being the order of the solutions and An (x) a family of continuous functions. It is proved that, with a convenient normalization of the solutions, $T_{\pm1}(z)=z\pm \mbox{\rm sign}(d)\arctan(y_n (x(z))/y_{n\pm 1}(x(z)))$ are globally convergent iterations with fixed points $z(x_n^{(i)})$, $x_n^{(i)}$ being the zeros of yn(x); d is one of the coefficients in the DDEs and z(x) is a primitive of d. The structure of the DDEs is also used to set global bounds on the differences between adjacent zeros of functions of consecutive orders and to find iteration steps which guarantee that all the zeros inside a given interval can be found with certainty. As an illustration, we describe how to implement this scheme for the calculation of the zeros of arbitrary solutions of the Bessel, Coulomb, Legendre, Hermite, and Laguerre equations.