WKB approximation and Krall-Type orthogonal polynomials

Abstract

We give an uni ed approach to the Krall-type polynomials orthogonal with respect to a positive measure consisting of an absolutely continuous one \perturbed" by the addition of one or more delta Dirac functions. Some examples studied by di erent authors are considered from an unique point of view. Also some properties of the Krall polynomials are studied. The three-term recurrence relation is calculated explicitly, as well as some asymptotic formulas. With special emphasis will be considered the second order di erential equations that such polynomials satisfy which allows us to obtain the central moments and the WKB approximation of the distribution of zeros. Some examples coming from quadratic transformation polynomial mappings and tridiagonal periodic matrices are also studied.