I am trying to find probability generating function for $P\left\{ X > n+1\right\} $.
Let X be a random variable assuming the values $0, 1, 2, ...$. The notation both for the distribution of $X$ and for it's tails are $P\left\{ X = j\right\} = p_j$, $P\left\{ X > j\right\} = q_j$. So the generating functions of the sequences $\{p_j\}$ and $\{q_j\}$ are $P(s) = p_0 + p_1s+ p_2s^2 + p_3s^3+...$ and $Q(s) = q_0 + q_1s+ q_2s^2 + q_3s^3+...$
As $P(1) = 1$, the series for $P(s)$ converges absolutely at least for $-1 \leq s \leq 1$. The coefficients of $Q(s)$ are less than unity, and so the series for $Q(s)$ converges at least in the open interval $-1 < s < 1$.
Also for $-1 < s < 1$ there is a known identity which provides the relation $Q\left( s\right) =\dfrac {1-P\left( s\right) } {1-s}$
In most of similar problems the approach i have been taking is to convert the desired probability into some form of an algebraic equation of $P\left\{ X = j\right\}$, $P\left\{ X > j\right\}$ and may be 1. Then substitute in one of the known generating function and solve for an expression of the generating function of the desired probability.
I am unsure this time this idea is working as i can n't seem to shake off of the tail due to +1 on the n.
$P\left\{ X > n+1\right\} = 1-P\left\{ X \leq n+1\right\} $ $=1 -P\left\{ X = n+1\right\}-P\left\{ X < n+1\right\}$ I am unsure if should further expand $P\left\{ X < n+1\right\}$ part as it would keep unfolding recursively.
Any help would be much appreciated.