Asymptotic estimates for Laguerre polynomials

We give a brief summary of recent results concerning the asymptotic behaviour of the Laguerre polynomialsL _n ⁽⁾ (x). First we summarize the results of a paper of Frenzen and Wong in whichn and >–1 is fixed. Two different expansions are needed in that case, one with aJ-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which is not necessarily fixed. These results follow from papers of Dunster and Olver, who considered the expansion of Whittaker functions. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeros of the approximant with the related zeros of the Laguerre polynomial.