SOME NEW ASYMPTOTIC PROPERTIES FOR THE ZEROS OF JACOBI, LAGUERRE, ANDHERMITE-POLYNOMIALS

For the generalized Jacobi, Laguerre, and Hermite polynomials P-n((alp ha n,beta n))(x), L(n)((alpha n))(X), H-n((gamma n))(X) the limit dist ributions 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 sp ecial properties of the sequences in the corresponding recurrence form ulas. The results are used to give second-order approximations for the largest and smallest zero which improve (and generalize) the limit st atements in a paper by Moak, Saff, and Varga [11].