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

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Abstract

Some of the work on the construction of inequalities and asymptotic approximations for the zeros λn,k(α), k=1,2,…,n, of the Laguerre polynomial Ln(α)(x) as ν=4n+2α+2→∞, is reviewed and discussed. The cases when one or both parameters n and α unrestrictedly diverge are considered. Two new uniform asymptotic representations are presented: the first involves the positive zeros of the Bessel function Jα(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.

MSC

Primary 33C45
65D20
Secondary 33C10
41A60

Keywords

Bessel functions
Airy functions
Whittaker functions, Hermite polynomials
Liouville–Green transform
Uniform approximation

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