## Characterizations of generalized Hermite and sieved ultraspherical polynomials.(English)Zbl 0863.33006

Consider generalized Hermite polynomials that are orthogonal with respect to the measure $$|x^\gamma |\exp (-x^2) dx (\gamma> -1)$$. The author develops two characterizations of these polynomials by utilizing the “reversing property” of the coefficients in the corresponding three term recurrence relations. The development contrasts with previous work which relied on explicit representations derived from Laguerre and Jacobi polynomials. The analysis is also applied to polynomials that are orthogonal with respect to the measure $$|x^\gamma |(1- x^2)^{1/2} dx$$. Generalized sieved ultraspherical polynomials of the first and second kinds are treated similarly. As applications, the asymptotic limit distribution for the zeros is determined when the degree of the parameters approaches infinity with the same order. The motivation arises from the study of optimal experimental designs.

### MSC:

 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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