---
tag: 6YD7S3C7W7
kind: definition
name: p-adic gamma function
msc: ["11S80", "33B15"]
defines: 84MVNJCYR4
uses: [V6TJAFZ4MD, 0XWG0JGDFW]
sources:
  - url: https://en.wikipedia.org/wiki/P-adic_gamma_function
---

## Definition

Let $p$ be a prime number and let $\Z_p$ be the ring of [[p-adic integers|V6TJAFZ4MD]].
For a positive integer $n$, put
$$\Gamma_p(n) = (-1)^n \prod_{\substack{0 < i < n \\ p \nmid i}} i.$$
The *$p$-adic gamma function* is the unique continuous function $\Gamma_p \colon \Z_p \to \Z_p$ that takes these values at the positive integers.

## Remarks

The positive integers are dense in $\Z_p$, so such a continuous extension is unique.
It exists because $\Gamma_p(m)$ and $\Gamma_p(n)$ are $p$-adically close when $m$ and $n$ are; the proof uses a generalisation of Wilson's theorem.
The values of $\Gamma_p$ are units of $\Z_p$.

Example: for $p = 3$, $\Gamma_3(1) = -1$, $\Gamma_3(2) = 1$, $\Gamma_3(3) = -2$, $\Gamma_3(4) = 2$ and $\Gamma_3(5) = -8$.
Each value follows from the previous one by the rule
$$\Gamma_p(x + 1) = \begin{cases} -x\, \Gamma_p(x) & \text{if } x \in \Z_p^\times, \\ -\Gamma_p(x) & \text{if } x \in p\Z_p, \end{cases}$$
which holds for all $x \in \Z_p$; here $\Gamma_p(0) = 1$.

Non-example: the [[factorial|0XWG0JGDFW]] $n \mapsto n!$ has no continuous extension to $\Z_p$.
Indeed $1 + p^k \to 1$ in $\Z_p$ as $k \to \infty$, but $(1 + p^k)! = (1 + p^k) \cdot (p^k)!$ is divisible by $p^k$ and so tends to $0$, while $1! = 1$.