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Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials

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2012-03-01
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In this paper we state and prove some properties of the zeros of exceptional Jacobi and Laguerre polynomials. Generically, the zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between zeros of consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A Heine-Mehler type formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m. We also describe the location and the asymptotic behaviour of the m exceptional zeros, which converge for large n to fixed values.
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zeros, outer relative asymptotics, Heine-Mehler formulas, Sturm-Liouville problems, algebraic Darboux transformations, exceptional orthogonal polynomials
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Journal of Mathematical Analysis and Applications, 399 (Issue 2), pp. 480-495