First-return generating function of the simple symmetric random walk (${\mathbb{E}}_0[s^T]=1-\sqrt{1-s^2}$)

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.