Some new asymptotic properties for the zeros of Jacobi, Laguerre and Hermite polynomials

Abstract

For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{({\alpha }_n,{\beta }_n)}(x),L_n^{({\alpha }_n)}(x),$\break $H_n^{({\gamma }_n)}(x)$ the limit distributions of the zeros are found, when the sequences ${\alpha }_n$ or ${\beta }_n$ tend to infinity with a larger order than $n$. The derivation uses special properties of the sequences in the corresponding recurrence formulae. The results are used to give second order approximations for the largest and smallest zero which improve (and generalize) the limit statements in a paper of Moak, Saff and Varga [11].