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

Abstract

Some of the work on the construction of inequalities and asymptotic approximations for the zeros λ n, k ( α) , k=1,2,…, n, of the Laguerre polynomial L n ( α) ( x) as ν=4 n+2 α+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 J α ( 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.