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On polynomial eigenfunctions for a class of differential operators. (English) Zbl 1016.34083
The authors study the distribution of zeros of polynomial eigenfuctions of differential operators with polynomial coefficients \(Q_j\): \(T_Q(f) = \sum_{j=0}^k Q_j \partial^j f/ \partial z^j\). They prove that the root measures of the monic eigenpolynomials (of degree \(n\)) of the operator \(T_Q\) converge weakly to a probability measure when \(n\to\infty\). It is shown that the limiting measure is determined by the leading polynomial \(Q_k\). Properties of the limiting measure (structure of the support, relation to \(Q_k\)) are given.

MSC:
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
47E05 General theory of ordinary differential operators
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
30C10 Polynomials and rational functions of one complex variable
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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