Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 4 down vote favorite

Does anyone know how to derive a formula for the coefficients.

That is if, $f(x)=\sum _{n=0}^{\infty } a_nx^n$ and $g(x)=\sum _{n=0}^{\infty } b_nx^n$

suppose the compostion is an analytic function, $h(x)=f(g(x))=\sum _{n=0}^{\infty } c_nx^n$

Is there an expression we can find for the coefficients $c_n$ in terms of $a_n$ and $b_n$? Can someone show me how its derived. I know we could substitute $g$ into $f$ and collect powers of $x$. But I believe a formula for general n may be written down.

share|improve this question

3 Answers

up vote 2 down vote accepted

There are (rather unwieldly) "closed-forms" in terms of Bell polynomials and other closely related combinatorial objects. However, if you are really interested in efficiently calculating compositions of power series then there are better algorithms, dating back at least to the work of Brent and Kung, from which you can find links to recent work in this area.

share|improve this answer
But these unwieldly formulas refer to exponential generating functions, do any results exist for standard generating functions? None the less can you point me to any references that derive such formula for the coefficients... Not sure where to look, books on combinatronics, analytic functions or ... ? thanks in advance. I'll look at some of the mentioned algorithms once i've investigated the formulas. I'm particularly curious about their derivation. – aukie Mar 5 '11 at 20:20
1  
@aukie: That and much more should be in the classic Schwatt: Operations with series – Gone Mar 5 '11 at 20:29
aplologies, but i'm finding it hard to source this reference, is there any others you could provide? – aukie Mar 5 '11 at 21:00
@auki: In general there's not much more that can be said beyond Faà di Bruno's formula and Bell polynomials and related results. You should be able to find such results in most combinatorics textbooks, e.g. the classic introduction by Riordan. Perhaps if you say more about your motivation then we could say more. – Gone Mar 5 '11 at 21:12
up vote 3 down vote

Please refer this link:

share|improve this answer
But the coefficients themselves in this post are infintite series ... is there no recurrent or closed form relationship between $a_n$, $b_n$ and $c_n$? so that the unkown coefficients can be computed... – aukie Mar 5 '11 at 19:27
up vote 2 down vote

There is a closed form, but it is kind of complicated: http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula

share|improve this answer
Interesting, so we can in terms of bell polynomials... do you know how this formula is derived ... or can you cite me any references? This doesn't directly answer my question since the series involved are exponential power series. but still i would be interested in seeing its proof. – aukie Mar 5 '11 at 20:00
1  
The Faa di Bruno formula does give explicit formulas for $c_n$ in terms of $a_n$ and $b_n$ (if you look at that wikipedia page it gives it at the beginning as a complicated but finite sum). I don't believe there's any simple formula. Steven Krantz's book "A primer of real-analytic functions" gives a proof of it, that's where I've seen it. – Zarrax Mar 5 '11 at 20:26

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.