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

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.