---
tag: A30TFC9Z92
kind: definition
name: Multiple gamma function
msc: ["33B15", "11M35"]
defines: FM0ZZS1PY2
uses: [G4SW7HJG88, 0M3YQJ81NB, 0H5SYBSED9]
sources:
  - url: https://en.wikipedia.org/wiki/Multiple_gamma_function
---

## Definition

Let $N \ge 1$ be an integer, let $a = (a_1, \ldots, a_N)$ with real $a_i > 0$, and let $w$ be a complex number with $\operatorname{Re} w > 0$.
The *Barnes zeta function* is
$$\zeta_N(s, w \mid a) = \sum_{n_1, \ldots, n_N \ge 0} \frac{1}{(w + n_1 a_1 + \cdots + n_N a_N)^s}, \qquad \operatorname{Re} s > N,$$
with the principal branch of the power.
It has a meromorphic [[analytic continuation|0M3YQJ81NB]] in $s$ to $\C$ that is holomorphic at $s = 0$.
The *multiple gamma function* of order $N$ is
$$\Gamma_N(w \mid a) = \exp\left(\left.\frac{\partial}{\partial s} \zeta_N(s, w \mid a)\right|_{s = 0}\right).$$

## Remarks

Example: for $N = 1$ and $a_1 = 1$, $\zeta_1(s, w \mid 1)$ is the [[Hurwitz zeta function|G4SW7HJG88]] $\zeta(s, w)$.
Lerch's formula $\frac{\partial}{\partial s}\zeta(s, w)\big|_{s=0} = \log \Gamma(w) - \frac12 \log 2\pi$ gives $\Gamma_1(w \mid 1) = \Gamma(w)/\sqrt{2\pi}$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].

The identity $\zeta_N(s, w \mid a) - \zeta_N(s, w + a_j \mid a) = \zeta_{N-1}(s, w \mid a^{(j)})$, where $a^{(j)}$ omits $a_j$, gives the functional equation
$$\Gamma_N(w + a_j \mid a) = \frac{\Gamma_N(w \mid a)}{\Gamma_{N-1}(w \mid a^{(j)})},$$
with the convention $\Gamma_0(w) = 1/w$.
For $N = 1$ it reads $\Gamma_1(w + 1 \mid 1) = w\, \Gamma_1(w \mid 1)$, the functional equation of $\Gamma$.

Non-example of a valid parameter: if $a_1 = 0$, each term is repeated for every value of $n_1$, the series diverges for every $s$, and $\Gamma_N$ is not defined.

Several normalisations are in use; they differ by factors of the form $\exp(\text{polynomial in } w)$.
The function $1/\Gamma_N(w \mid a)$ extends to an entire function of $w$.