Some properties of orthogonal polynomials satisfying fourth order differential equations

Abstract

Introduction.In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2-8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

Electrostatic interpretation of zeros of orthogonal polynomials

“…One direction of the statement (zeros of a polynomial solutions satisfy the system of equations) is a fairly straightforward computation and the outline of the argument can be found, for example, in [8] or, as a remark, in [11, §3]. Moreover, results in this direction can be obtained for much more general differential equations of higher order (see, for example, [4,5,6]). Arguments in the other direction seem to exist only in special cases, as in the work of Stieltjes, and are based on interpreting the system of equations as the critical point of an associated energy functional (see e.g.…”

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

“…The first is the electrostatic interpretation of the zeros of polynomials satisfying second order differential equations [15]- [17], 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 [8]- [11]. The formulas yielded by the electrostatic interpretation of the zeros of Bessel polynomials are used to find them numerically as it has been done previously with these and other sets of points [7]- [19]. Several sets of zeros are computed in this way and the sets of real and imaginary values are fitted by polynomials depending on the index k whose coefficients depend on n. Finally, it is found that the approximate expression for the kth zero of y n (x) is O(1/n 2 )-convergent to a limit point of the zeros of the Bessel polynomials.…”