From: Wilbert Dijkhof <w.j.dijkhof@student.utwente.nl>
Subject: Re: Sum ( x^(k^2)) where k =0,1,2...n ?
Date: Sun, 01 Oct 2000 13:59:45 +0200
Newsgroups: sci.math
Summary: [missing]

Rolf Kleinknecht wrote:
> 
> There is a result guessed by Euler and prooved by one of these
> theory-of-function guys in the 19th century for an infinite series:
> 
> ( (1+x)(1+x^2)(1+x^3)... )^-1 =
>         = 1 + SUM( j=1..INF: (-1)^j ( x^(j(3j-1)) - x^(j(3j+1)) )
> 
> or the like. ( I don't quite remember the correct signs etc.)
> 
> I presume there are some more formulas of this kind.
> 
> Yet for finite sums?
> 
> ro_kl@t-online.de

I think you are referring to "Euler's pentagonal number theorem"
which states:

(1+x)(1+x^2)(1+x^3)... = SUM( (-1)^j * x^(j(3j+1)/2), j=-INF..INF )

which is a special result of "Jacobi's triple-product identity"
which states:

product[ (1 + y*x^(2n-1)) * (1 + y^(-1)*x^(2n-1)) * (1 - x^(2n)) ,
n=-INF..INF]
= SUM( y^j * x^(j^2), j=-INF..INF )

Wilbert
==============================================================================

From: Wilbert Dijkhof <w.j.dijkhof@student.utwente.nl>
Subject: Re: Sum ( x^(k^2)) where k =0,1,2...n ?
Date: Sun, 01 Oct 2000 19:52:15 +0200
Newsgroups: sci.math

Wilbert Dijkhof wrote:
> 
> 
> I think you are referring to "Euler's pentagonal number theorem"
> which states:
> 
> (1+x)(1+x^2)(1+x^3)... = SUM( (-1)^j * x^(j(3j+1)/2), j=-INF..INF )

I meant (1-x)(1-x^2)(1-x^3)... = SUM( ... ) :)
 
Wilber