I have just came across a weird definition for the probability generating function of a random variable $N$ that denotes the integer value for the $n^{\mathrm{th}}$ toss on which the coin turned out to be heads.
First of all, I always thought that the lower bound in the sum for the definition of probability generating function is zero, i.e.:
$$G(x) = \sum_{\color{red} {i=0}}^{\infty}x^i P(X=x_i)$$
I also thought the the toss on which heads is to turn up is independent of the previous tosses and as such at each toss, the probability is always constant ... i.e. a $1/2$. Now the definition of the pgf that I just saw is:
$$G(x) = \sum_{\color{red} {i=1}}^{\infty} x^i \left( \frac{1}{2}\right)^i$$
This can be seen here: STEP III,2011,Q12 (post number $11$).
I would imagine that the pgf has to be:
$$G(x) = \sum_{\color{red} {i=0}}^{\infty} x^i \left( \frac{1}{2}\right)$$