I had first wondered about a special case of, how to calculate the number of possibilities so that the sum of eyes of $n$ thrown dices equals a certain value $c$.
After some time thinking, I decided to google and found on mathworld the following equations describing the general case of my initial problem. For simplification I have copied the interesting lines and posted them here:
\begin{align} f(x) &= (x+x^1 + \dots + x^s )^n \tag 1 \\ & = x^n {\left( \sum\limits_{i=0}^{s-1}x^i\right)^n} \tag 2 \\ & = x^n \left(\frac{1-x^s}{1-x}\right)^n \tag 3 \\ & = x^n (1-x^s)^n \left(\frac{1}{1-x}\right)^n \tag 4 \\ & = x^n \sum\limits_{k=0}^{n} (-1)^k \binom{n}{k} x^{sk} \sum\limits_{l = 0}^{\infty} \binom{n+l-1}{l} x^l \tag 5 \end{align}
The steps down from $1-4$ are not troublesome at all. From step $4$ to $5$ one uses the binominal theorem and then its generalized version.
Using the gerenalized version of the binominal theorem however requires the condition: $x < |1| $
It had not been stated on mathworld, but obviously assumed to do that step. T My question now: How do we assume that and why? What do the $x$ mean in this generating function, if one has to add additional condition on them?
My guess:
Obviously the exponents in $x, \dots, x^s$ are simply the representation of the numbers $1$ to $s$ on our dice and that if the dice is fear, then the whole expression $x, \dots, x^s$ represent the resprective likelihoods that a certain result appears in a throw and then making their coefficients of the number of possible combinations to get to that result, well, of course the likelihoods are $\in [0,1]$ . My problem however here is that the generating function here interemixes arithmetic rules with representations as actually just the exponents and the coefficient are interesting.
That's exactly what generating functions are about: exponents represent the values of some parameter and the coefficient represents "how many cases". The "x" is just a placeholder at which to attach the exponent and the quantity. The method of generating functions works for problems that "fit" with power series arithmetic.