# Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+n_sq^s$$ be its decomposition in basis $q$. Then, Carlitz puts for the $n$-th factorial of $\mathbb F_q[T]$ $$g_n=\prod_{i=0}^sD_i^{n_i}.$$ For a prime $P$ of $\mathbb F_q[T]$, denote by $\mathbb Fq(T)_P$ be the completion of $\mathbb F_q(T)$ for the $P$-adic valuation. The fact that $\lim_{n\to+\infty}\limits v_P(g_n)=+\infty$ implies that for for every $z\in\mathbb F_q[T]_P$, the series $g(z)=\sum_{n\ge0}g_nz^n$ converges. But I did not manage to prove that the series converges for any $z\in\mathbb F_q(T)_P$. Does anyone have an idea to show that?