asympt(hypergeom(...)) problem

[John Harper]

Five years ago I complained to Waterloo Maple Inc. that Maple V.5
couldn't do asymptotic expansions of hypergeometric functions, though V.4
could if they reduced to more elementary functions. This problem has
still not been fixed! Asymptotic expansions of pFq and some more general
functions have of course long been known (Wright, E.M., Proc.Lond.Math.Soc.
46, 389-408, 1940). They need the sort of complicated algebra that I had
thought Maple was designed for. In the example below the hypergeometric
function can be simplified, though not by asympt, but Maple 9.5 fails even
when helped by a simplify command to give an O(something) error estimate.

|\^/| Maple 9.5 (IBM INTEL LINUX)
._|\| |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2004
\ MAPLE / All rights reserved. Maple is a trademark of
<____ ____> Waterloo Maple Inc.
| Type ? for help.
> asympt(hypergeom([1],[3/2],x),x);
Error, (in asympt) unable to compute series
> simplify(hypergeom([1],[3/2],x));
1/2 1/2
exp(x) Pi erf(x )
1/2 ----------------------
1/2
x

> asympt(%,x);
1/2 1/2 1 1 3 15
1/2 Pi (1/x) exp(x) - --- + ---- - ---- + -----
2 x 2 3 4
4 x 8 x 16 x

John Harper, School of Mathematical and Computing Sciences,
Victoria University, PO Box 600, Wellington, New Zealand
e-mail john....@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045