How to find an explicit formula from a generating function

Question

Having read on Wikipedia about generating functions, I still don't understand what I can do with a generating function.

For example, if I want to compute the first spread polynomial, how can I do that from a generating function, such as these (taken from the Wikipedia article):

$$ \sum_{n=1}^\infty S_n(s)x^n = {sx(1+x) \over (1-x)^3 + 4sx(1-x)}. $$

An exponential one:

$$ \sum_{n=1}^\infty {S_n(s)\over n!} x^n = {1 \over 2} e^x \left [ 1-e^{-2sx} \cos\left (2x \sqrt{s(1-s)}\right )\right ] $$

Answer 1

If you know what Taylor expansion is, then you should know that $S_n(s)$ in e.g.f. is just its $n$th derivative at zero. For the ordinary g.f., it's almost the same, except that you have to divide the derivative by $n!$.

Anyway, if you want to do more with g.f. than this, I strongly advise you to at least look at this wikipedia page and perhaps even into the generatingfunctionology book, which in my opinion does a pretty good job of introducing g.f. to a beginner.

Answer 2

HINT $\quad\quad $ With $\rm\ \ \alpha\ =\ 1 - 2\ s + 2\ \sqrt{s^2 - s}\ \ $ we have

$$ \rm \frac{s\ x\ (1+x)}{(1-x)^3 + 4\ s\ x\ (1-x)}\ =\ \frac{\alpha}{4\ (x-\alpha)} + \frac{\alpha'}{4\ (x-\alpha')} - \frac{1}{2\ (x-1)} $$

Answer 3

Here it's easier to work with the exponential generating function. You just need to find the Taylor series of each term on the RHS. You already know the Taylor series of one term. To find the Taylor series of the other term, use the fact that $\cos t = \frac{e^{it} + e^{-it}}{2}$.