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

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

1994-06-07

arxiv.org/abs/math/9406224v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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].

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