Asymptotics and bounds for the zeros of Laguerre polynomials: a survey

[摘要] Some of the work on the construction of inequalities and asymptotic approximations for the zeros lambda(n,k),((alpha)) k = 1,2,...,n, of the Laguerre polynomial L-n((alpha))(x) as v = 4n + 2alpha + 2 --> infinity, is reviewed and discussed. The cases when one or both parameters n and a unrestrictedly diverge are considered. Two new uniform asymptotic representations are presented: the first involves the positive zeros of the Bessel function J(alpha)(x), and the second is in terms of the zeros of the Airy function Ai(x). They hold for k = 1, 2, . . ., [qn] and for k = [pn], [pn] + 1,. . .,n respectively, where p and q are fixed numbers in the interval (0, 1). Numerical results and comparisons are provided which favorably justify the consideration of the new approximations formulas. (C) 2001 Elsevier Science B.V. All rights reserved.

[发布日期] 2002-07-01

[效力级别]  Proceedings Paper

[关键词] Bessel functions;airy functions;Whittaker functions;hermite polynomials;Liouville-Green transform;uniform approximation