Uniform Approximations for the Zeros of Laguerre Polynomials

Abstract

In this paper we obtain two asymptotic formulas for the zeros $\lambda _{n,k}^{(\alpha )},k = 1,2, \ldots ,n,$ of the Laguerre polynomials $L_n^{(\alpha )}(x)$, as n → ∞ and α is fixed. These formulas are in terms of the zeros of the Bessel function J (x) and in terms of the zeros of the Airy function Ai(χ). 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).