From: Paul.Zimmermann@loria.fr (Paul Zimmermann)
Subject: Re: Taylor Coefficients of an algebraic function
Date: 19 May 00 17:10:58 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

   Date:         Thu, 18 May 2000 19:12:33 -0400
   From: Alf van der Poorten <alf@ics.mq.edu.au>

   The best way to evaluate the Taylor coefficients of an algebraic function $y=y(x)$, thus satisfying a polynomial equation $f(x,y)=0$, is first to obtain a linear differential equation satisfied by $y$ and thence a $P$-recurrence on its Taylor coefficients.

   I've forgotten how to to obtain, in an elegant way, a linear differential equation satisfied by an algebraic function $y$ in terms of its defining polynomial $f$ and am seeking reminders/references.

One way is to use the Maple gfun package:

> with(share):
See ?share and ?share,contents for information about the share library
> with(gfun):
> f := y=1+x*y^2:
> deq := algeqtodiffeq(f, y(x));
                                                    2  /d      \
              deq := 1 + (-1 + 2 x) y(x) + (-x + 4 x ) |-- y(x)|
                                                       \dx     /

> rec := diffeqtorec(deq, y(x), a(n));
             rec := {(2 + 4 n) a(n) + (-2 - n) a(n + 1), a(0) = 1}

> foo := rectoproc(rec, a(n)):
> seq(foo(n), n=0..13);
    1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900
> foo(100);
           896519947090131496687170070074100632420837521538745909320

The algorithm implemented by algeqtodiffeq is due to Comtet [Comtet64].
Gfun contains other useful functions, see [SaZi94].

Hope this helps,
Paul Zimmermann

@article{Comtet64,
  author =      "L. Comtet",
  title =       "Calcul pratique des coefficients de {T}aylor d'une fonction
                alg{\'e}brique",
  journal =     "Enseignement Math.",
  year =        "1964",
  volume =      10,
  pages =       "267--270"
}

@article{SaZi94,
  author =       "B. Salvy and P. Zimmermann",
  title =        "Gfun: A {M}aple Package for the Manipulation of
                  Generating and Holonomic Functions in One Variable",
  journal =      toms,
  volume =       20,
  number =       2,
  year =         1994,
  month =        jun,
  pages =        "163--177"
}