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

Abstract

For the generalized Jacobi, Laguerre, and Hermite polynomials $$P_n^{\left( {\alpha _n ,\beta _n } \right)} \left( x \right),L_n^{\left( {\alpha _n } \right)} \left( x \right),H_n^{\left( {\gamma _n } \right)} \left( x \right)$$ , the limit distributions of the zeros are found, when the sequences α n or β n tend to infinity with a larger order thann. The derivation uses special properties of the sequences in the corresponding recurrence formulas. 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 by Moak, Saff, and Varga [11].