W. Gawronski, Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials L n and Hermite polynomials H n , Analysis 13 (1993), 29--67.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....variable) can be used to obtain asymptotic zero distributions of many systems of orthogonal polynomials for which the recurrence coecients are explicitly known. In this way one may rederive, for example, the results on zero distributions of varying Hermite and Laguerre polynomials given in [1, 4, 8, 9, 10]. 4.3 Hahn polynomials Recently, classical orthogonal polynomials of a discrete variable have been studied from the point of view of extremal problems in potential theory. In order to describe the asymptotic zero distribution, Rakhmanov [29] found that the minimal energy problem stated in ....

W. Gawronski, Strong asymptotics and the asymptotic zero distribution of

....P.O. Box 94079, 1090 GB Amsterdam (NL) Kruislaan 413, 1098 SJ Amsterdam (NL) Telephone 31 20 592 9333 Telefax 31 20 592 4199 1 Uniform Asymptotic Expansions of Integrals: A Selection of Problems Paper presented at the Conference in honour of Thomas Jan Stieltjes Jr. 1856 1894) October 31 November 4, 1994, Delft University of Technology, The Netherlands N. M. Temme CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands e mail: nicot cwi.nl Abstract On the occasion of the conference we mention examples of Stieltjes work on asymptotics of special functions. The remaining part ....

....1 360z 3 1 1260z 5 : as z 1; j argzj . More precisely ln Gamma(z) z ln z Gamma z 1 2 ln(2=z) N Gamma1 X n=1 B 2n 2n (2n Gamma 1) 1 z 2n Gamma1 RN (z) 1 ) T.J. Stieltjes (1889) Sur la d eveloppement de log Gamma(a) J. Math. Paris, s er. 4, 5, 425 444. Collected papers (1993), Vol. 2, Springer Verlag, Berlin, 215 234. 2 where B n are the Bernoulli numbers and RN (z) O 1 z 2N Gamma1 ; z 1; j argzj : Stieltjes showed that: jRN (z)j jB 2N j 2N(2N Gamma 1) 1 cos 2N ( 1 2 ) 1 jzj 2N Gamma1 where = arg z 2 ( Gamma; A proof of this result ....

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Gawronski, W. (1993), Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials L an+ff n and Hermite polynomials H an+ff n , Analysis, 13, 29--67.

....variable) can be used to obtain asymptotic zero distributions of many systems of orthogonal polynomials for which the recurrence coefficients are explicitly known. In this way one may rederive, for example, the results on zero distributions of varying Hermite and Laguerre polynomials given in [2, 4, 8, 9, 10]. 4.3 Hahn polynomials Recently, classical orthogonal polynomials of a discrete variable have been studied from the point of view of extremal problems in potential theory. In order to describe the asymptotic zero distribution, Rakhmanov [29] found that the minimal energy problem stated in ....

W. Gawronski, Strong asymptotics and the asymptotic zero distribution of Laguerre

....H n should behave like its classical companion H n if the parameter N runs far ahead of the degree n. This speculation is made precise by the relation (1. 11) above saying that the limit distribution of the zeros for H n in this case is the same as for the standard Hermite polynomials H n [9]. In view of the weak asymptotics it is natural to ask for strong asymptotics; that is, for asymptotic forms of the rescaled polynomials H n (c n z) c n ) being a suitable sequence of positive numbers. A direct approach to answer this question is to start from (1.3) which gives the integral ....

W. Gawronski, Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials L n and Hermite polynomials H n , Analysis 13 (1993), 29--67.

....H N n should behave like its classical companion H n if the parameter N runs far ahead of the degree n. This speculation is made precise by the relation (1. 11) above saying that the limit distribution of the zeros for H N n in this case is the same as for the standard Hermite polynomials H n [9]. In view of the weak asymptotics it is natural to ask for strong asymptotics that is for asymptotic forms of the rescaled polynomials H N n (c n z) c n ) being a suitable sequence of positive numbers. A direct approach to answer this question is to start from (1.3) which gives the integral ....

W. Gawronski, Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials

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