 Solution

The first return time ${\tau }_1$ has distribution ${\mathbb{P}}_1({\tau }_1=k)={\alpha }_k$. It is finite almost surely, so state one is recurrent. It is positive recurrent precisely when its mean return time is finite:By irreducibility the same recurrence class contains every state. Thus the condition is finiteness of the expected value of the positive-integer distribution $({\alpha }_k)$. If that mean is infinite, the chain is null recurrent, not transient.