I'm reading a paper on Branching Processes and the Theory of Epidemics, and the fourth page (p. 262 of the book) the author refers to "the well-known formulas that gives the relation between the generating functions of a sequence and the sequence of its 'tails.'" Well, it's not well-known to me, and when I tried searching for it my results were all clogged up with probability generating functions for heads/tails coin tosses.
Any ideas? My REU group members contributed primarily obscene jokes, so.
You don’t actually need to know exactly to what he’s referring there, since the calculation that he needs is given in full detail at $(7)$. What he does there is a two-variable version of the following one-variable derivation of the relationship that he has in mind.
Let $\langle a_k:k\in\Bbb N\rangle$ be a summable sequence with generating function $A(x)$, and for $n\in\Bbb N$ let $b_n=\sum_{k>n}a_n$. Then
$$\begin{align*} G(x)&=\sum_{n\ge 0}b_nx^n=\sum_{n\ge 0}\sum_{k>n}a_kx^n\\\\ &=\sum_{k\ge 1}\sum_{0\le n<k}a_kx^n\\\\ &=\sum_{k\ge 1}a_k\sum_{0\le n<k}x^n\\\\ &=\sum_{k\ge 1}a_k\cdot\frac{1-x^k}{1-x}\\\\ &=\frac1{1-x}\left(\sum_{k\ge 1}a_k-\sum_{k\ge 1}a_kx^k\right)\\\\ &=\frac1{1-x}\big(b_0-A(x)\big)\;. \end{align*}$$