What does $\delta_n$ refer to in "distribution of $T_k$", when defining mean recurrence time?
Particularly
$$T_k:= \inf \{ n \geq 1: f_n=k\}$$
then my notes say:
$$\mathbb{P}(f_n=k, f_{n-1} \not=k, ..., f_1 \not=k | f_0=k)= f_{kk}^{(n)}$$
and that the distribution of $T_k$ is
$$\mathbb{P}_{T_k} = \sum_{n=1}^{\infty} f_{kk}^{(n)} \delta_n$$
Is this some "standard" $\delta$?
$\delta_n$ is a standard notation for the probability distribution that puts all its mass at the point $n$.
So your second displayed formula implies that $P(T_k=n) = f_{kk}^{(n)}$.