The discrete analogue of Laplace's method
We give a justification of the discrete analogue of Laplace's method applied to the asymptotic estimation of sums consisting of positive terms. The case considered is the series related to the hypergeometric function (p)F(q-1) (X) (with q >= p + 1) as x -> +infinity discussed by Stokes [G.G. Stokes, Note on the determination of arbitrary constants which appear as multipliers of semi-convergent series, Proc. Camb. Phil. Soc. 6 (1889) 362-366]. Two examples are given in which it is shown how higher order terms in the asymptotic expansion may be derived by this procedure. (C) 2011 Elsevier Ltd. All rights reserved.
laplace's method sums asymptotic expansion