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

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.The research of the first author (DGU) has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, under grant MTM2009- 06973. The work of the second author (FM) has been supported by Direccioón General de Investigaci ón, Ministerio de Ciencia e Innovación of Spain, grant MTM2009-12740-C03-01. The research of the third author (RM) was supported in part by NSERC grant RGPIN-228057-2009