# generating function problem

• Dec 11th 2008, 12:15 AM
kkkkkk
generating function problem
If the generating function is (1-x^200)(1-x^240)/((1-x^10)(1-x^20)(1-x^25)). Then expand all the terms through x^100. How can I make it in the form of a recurrence relation hn. The initial conditions are given.
• Dec 13th 2008, 02:34 PM
awkward
Quote:

Originally Posted by kkkkkk
If the generating function is (1-x^200)(1-x^240)/((1-x^10)(1-x^20)(1-x^25)). Then expand all the terms through x^100. How can I make it in the form of a recurrence relation hn. The initial conditions are given.

The generating function is

$(1-x^{200})(1-x^{240})(1-x^{10})^{-1}(1-x^{20})^{-1}(1-x^{25})^{-1}$
$=(1-x^{200})(1-x^{240})\sum_{i=0}^\infty x^{10i} \sum_{j=0}^\infty x^{20j} \sum_{k=0}^\infty x^{25k}$

The first two factors, $=(1-x^{200})(1-x^{240})$, don't make any contribution to $x^n$ with $n < 200$, so all you have to do is expand enough of

$\sum_{i=0}^\infty x^{10i} \sum_{j=0}^\infty x^{20j} \sum_{k=0}^\infty x^{25k}$

to get the powers of $x$ up to $x^{100}$.

I would start by expanding
$\sum_{j=0}^5 x^{20j} \sum_{k=0}^4 x^{25k}$
up to $x^{100}$.

I don't know enough about what you want in the way of a recurrence to help you with that part of the problem.