First-return generating function of the simple symmetric random walk
= First-return generating function of the simple symmetric random walk
{title2=$\mathbb E_0[s^T]=1-\sqrt{1-s^2}$}
For the <simple symmetric random walk> on $\mathbb Z$, the <renewal equation> relates first-return and return <probability generating functions>. The central-binomial return series is $U(s)=(1-s^2)^{-1/2}$, giving $F(s)=1-U(s)^{-1}$. Its value tends to one at $s=1$, but its derivative diverges there: the walk has <null recurrent states>.