---
tag: 7E2JJWFF00
kind: concept
name: Barnes G-function
msc: ["33B15"]
sources:
  - url: https://en.wikipedia.org/wiki/Barnes_G-function
same_as:
  wikidata: Q808463
  wikipedia: Barnes G-function
---

## Summary

The Barnes G-function is an [[entire function|0G4F0XB9X2]] $G$ with $G(1) = 1$ and $G(z+1) = \Gamma(z)\, G(z)$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
It plays for the gamma function the role that the gamma function plays for the identity function.
At positive integers it gives the superfactorials $G(n) = 0!\,1!\cdots(n-2)!$.
It is, up to an elementary factor, a [[multiple gamma function|FM0ZZS1PY2]] of order two, and it appears in random matrix theory and in determinant formulas.

## Explanations

### L1 human

The factorial multiplies the numbers $1, 2, \ldots, n$.
The superfactorial multiplies the factorials $0!, 1!, \ldots$ in the same way.
The Barnes G-function fills in values of the superfactorial between the whole numbers, as the gamma function does for the factorial.
For example, $G(5) = 0! \cdot 1! \cdot 2! \cdot 3! = 12$.