Let $p>2$ be a prime. For $n \in \mathbb{Z}^+$ we can define
\begin{equation}
F(n) = (-1)^n \prod\_{1<i<n, p\nmid i} i.
\end{equation}
Since $\mathbb{Z}$ is dense in $\mathbb{Z}\_p$, we can extend $F$ uniquely to a continuous function on $\mathbb{Z}\_p$, which is the $p$-adic gamma function $\Gamma\_p$.


Is there any good way to explicitly compute $\Gamma\_p(a/b)$ where $a,b \in \mathbb{Z}$ such that $a/b \in \mathbb{Z}\_p^{\times}$? Specifically, I'm looking at $\Gamma\_5(4/11)$.


A bad idea is to evaluate $F(a\_n)$, where $a\_n$ are the partial sums of the $5$-adic expansion of $4/11$. I don't think this would work since $F(a\_n)$ becomes too difficult to compute the further you go in the expansion.


I'm aware that some values can be computed via the [Gross-Koblitz formula](https://en.wikipedia.org/wiki/Gross%E2%80%93Koblitz_formula), but I don't know how or if I can apply it to my situation.


Any help is appreciated!


**Edit:** As Henri Cohen points out, you can get Sage to do this:



```
[Input] Sage: R=Zp(5)
        Sage: x = R(4/11)
        Sage: x.gamma('pari')
[Output]      1 + 3*5 + 4*5^3 + 4*5^4 + 4*5^6 + 3*5^7 + 2*5^8 + 4*5^9 
              + 4*5^10 + 5^11 + 4*5^13 + 5^15 + 5^16 + 3*5^17 + 2*5^18 
              + 5^19 + O(5^20)

```


Computing the $p$-adic gamma function $\Gamma_p$
The $p$-adic gamma function is implemented in Pari/GP (hence available in Sage).
The algorithm used is due to Fernando Rodriguez-Villegas (but probably predates him) and is explained in detail both in his book on experimental mathematics published at Oxford, and in my Springer GTM 240.


