From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: Power Series With Binomials
Date: Tue, 07 Sep 1999 09:14:21 GMT
Newsgroups: sci.math
Keywords: Application of Lagrange inversion formula

In article <erh4lvu9oix3@forum.swarthmore.edu>,
  qqquet@hotbot.com (Leroy Quet) wrote:
> I know that sum_{m=0}^infinity[binomial(2m,m)x^m]=
> 1/sqrt(1-4x).
> But,in general, what is sum_{m=0}^infinity[binomial(rm,m)x^m]?
> (r is a positive integer.)

Consider the equation x = f(x) - f(x)^r. By the Lagrange inversion
formula this has a unique power series solution f(x) with f(0) = 0
and this is

f(x) = sum_{k=0}^infinity binomial(rk,k) x^{1+k(r-1)}/[1+k(r-1)].

Hence f'(x) = sum_{k=0}^infinity binomial(rk,k) x^{k(r-1)}.

Now f'(x)[1 - rf(x)^{r-1}] = 1 so f'(x) = 1/[1 - rf(x)^{r-1}].
Should we think the effort worthwhile we could get a degree
r equation for f' from that for f.

--
Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "They did not have proper palms at home in Exeter."
    Peter Carey, _Oscar and Lucinda_


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