Properties of zeros of orthogonal polynomials and related functions

Abstract

There are many results in the literature on orthogonal polynomials concerning the way in which the zeros of a polynomial change as a parameter changes. For the classical polynomials there are results due to Markoff (1886) and Stieltjes (1886) on the monotonic variation of zeros of Jacobi and Gegenbauer (ultraspherical) polynomials with respect to parameters. Our purpose here is to describe some recent results concerning formulas for the derivatives of zeros with respect to a parameter. Such formulas and their consequences are fairly well-developed for Bessel functions (a limiting case of some of the classical orthogonal polynomials) and have recently been explored for some orthogonal polynomials and related functions. Two general methods will be described. The first, use extensively by Ismail, the author and others, is based on the Hellmann-Feynman theorem which gives the derivative of an eigenvalue with respect to a parameter. It can be applied in either a differential equations or a recurrence relations setting. The second method, developed in recent work of Elbert and the author, and applicable to suitable solutions of second-order ordinary linear differential equations, originates in work of Richardson (1918). From this approach we obtain formulas for dc dλ , where c = c( λ) is a zero of the ultraspherical polynomial P ( λ) n ( x). These recover the known result that dc dλ < 0, λ > − 1 2 . We get similar results for certain Hermite functions. These results are closely related to Nicholson-type formulas, due to Durand (1975), for sums of squares of solutions of second-order linear ordinary differential equations. Nicholson-type formulas are named after the corresponding formulas for Bessel functions due to Nicholson.