Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials Articles uri icon

publication date

  • March 2012

start page

  • 480

end page

  • 495

volume

  • 399

international standard serial number (ISSN)

  • 0022-247X

electronic international standard serial number (EISSN)

  • 1096-0813

abstract

  • 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.