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

Abstract

For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{(α_n, β_n)} (x), L_n^{(α_n)} (x),$\break $H_n^{(γ_n)} (x)$ the limit distributions of the zeros are found, when the sequences $α_n$ or $β_n$ 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].