Some new asymptotic properties for the zeros of Jacobi,Laguerre, and Hermite polynomials |
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Authors: | H. Dette W. J. Studden |
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Affiliation: | 1. Institut für Mathematische Stochastik, Technische Universit?t Dresden, Mommsenstrasse 13, 01062, Dresden, Germany 2. Department of Statistics, Purdue University, 1399 Mathematical Sciences Building, 47907-1399, West Lafayette, Indiana, USA
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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]. |
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