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 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" }