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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.