Existence and uniqueness of the Iwasawa power series $G_p(\omega^j,T)$

Let $p$ be a prime, let $j$ be an even integer with $j \not\equiv 0 \pmod{p-1}$, and let $u \in \mathbb{Z}_p^\times$ be a topological generator of $1 + p\mathbb{Z}_p$. Then there is exactly one power series $G = \sum_n a_n T^n \in \mathbb{Z}_p[[T]]$ such that for every integer $k \ge 1$

where $B_{k,\chi} = p^{k-1}\sum_{a=1}^{p}\chi(a)B_k(a/p)$ is the generalized Bernoulli number of the character $\chi$ modulo $p$ and $\omega$ is the Teichmüller character.

In the source's notation, this is the statement that the Kubota–Leopoldt $p$-adic $L$-function $L_p(\omega^j, s)$, characterised by $L_p(\omega^j, 1-k) = -B_k(\omega^{j-k})/k$, is given by a power series with $\mathbb{Z}_p$-coefficients: $G_p(\omega^j, u^s - 1) = L_p(\omega^j, s)$. This makes $G_p(\omega^{1-i}, T)$ in the main conjecture well defined.

Formalization Note For $p = 2$ or $p = 3$ the hypotheses cannot be met (every even $j$ is divisible by $p-1$), so the statement is vacuous there.