Multiple gamma function

In mathematics, the multiple gamma function $Γ_{N}$ is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by Barnes (1901). At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in Barnes (1904).

Double gamma functions $Γ_{2}$ are closely related to the q-gamma function, and triple gamma functions $Γ_{3}$ are related to the elliptic gamma function.

Definition

For $ℜ a_{i} > 0$ , let

Γ_{N} (w ∣ a_{1}, \dots, a_{N}) = \exp ({\frac{\partial}{\partial s} ζ_{N} (s, w ∣ a_{1}, \dots, a_{N}) |}_{s = 0}),

where $ζ_{N}$ is the Barnes zeta function. (This differs by a constant from Barnes's original definition.)

Properties

Considered as a meromorphic function of $w$ , $Γ_{N} (w ∣ a_{1}, \dots, a_{N})$ has no zeros. It has poles at $w = - \sum_{i = 1}^{N} n_{i} a_{i}$ for non-negative integers $n_{i}$ . These poles are simple unless some of them coincide. Up to multiplication by the exponential of a polynomial, $Γ_{N} (w ∣ a_{1}, \dots, a_{N})$ is the unique meromorphic function of finite order with these zeros and poles.

$Γ_{0} (w ∣) = \frac{1}{w},$
$Γ_{1} (w ∣ a) = \frac{a^{a^{- 1} w - \frac{1}{2}}}{\sqrt{2 π}} Γ (a^{- 1} w),$
$Γ_{N} (w ∣ a_{1}, \dots, a_{N}) = Γ_{N - 1} (w ∣ a_{1}, \dots, a_{N - 1}) Γ_{N} (w + a_{N} ∣ a_{1}, \dots, a_{N}) .$

In the case of the double Gamma function, the asymptotic behaviour for $w \to \infty$ is known, and the leading factor is

Γ_{2} (w | a_{1}, a_{2}) \underset{w \to \infty}{\sim} w^{\frac{w^{2}}{2 a_{1} a_{2}}} for {\begin{matrix} \frac{a_{1}}{a_{2}} \in ℂ ∖ (- \infty, 0], \\ w \in ℂ ∖ (ℝ_{+} a_{1} + ℝ_{+} a_{2}) . \end{matrix}

Infinite product representation

The multiple gamma function has an infinite product representation that makes it manifest that it is meromorphic, and that also makes the positions of its poles manifest. In the case of the double gamma function, this representation is

Γ_{2} (w ∣ a_{1}, a_{2}) = \frac{e^{λ_{1} w + λ_{2} w^{2}}}{w} \prod_{\begin{matrix} (n_{1}, n_{2}) \in ℕ^{2} \\ (n_{1}, n_{2}) \neq (0, 0) \end{matrix}} \frac{e^{\frac{w}{n_{1} a_{1} + n_{2} a_{2}} - \frac{1}{2} \frac{w^{2}}{(n_{1} a_{1} + n_{2} a_{2})^{2}}}}{1 + \frac{w}{n_{1} a_{1} + n_{2} a_{2}}},

where we define the $w$ -independent coefficients

λ_{1} = - \underset{s = 1}{{Res}_{0}} ζ_{2} (s, 0 ∣ a_{1}, a_{2}),

λ_{2} = \frac{1}{2} \underset{s = 2}{{Res}_{0}} ζ_{2} (s, 0 ∣ a_{1}, a_{2}) + \frac{1}{2} \underset{s = 2}{{Res}_{1}} ζ_{2} (s, 0 ∣ a_{1}, a_{2}),

where $\underset{s = s_{0}}{{Res}_{n}} f (s) = \frac{1}{2 π i} \oint_{s_{0}} (s - s_{0})^{n - 1} f (s) d s$ is an $n$ -th order residue at $s_{0}$ .

Another representation as a product over $ℕ$ leads to an algorithm for numerically computing the double Gamma function.

A numerical implementation of the Barnes multiple gamma function is available in the Wolfram Function Repository.

Reduction to the Barnes G-function

The double gamma function with parameters $1, 1$ obeys the relations

Γ_{2} (w + 1 | 1, 1) = \frac{\sqrt{2 π}}{Γ (w)} Γ_{2} (w | 1, 1), Γ_{2} (1 | 1, 1) = \sqrt{2 π} .

It is related to the Barnes G-function by

Γ_{2} (w | α, α) = (2 π)^{\frac{w}{2 α}} α^{- \frac{w^{2}}{2 α^{2}} + \frac{w}{α} - 1} G (w / α)^{- 1} .

The double gamma function and conformal field theory

For $ℜ b > 0$ and $Q = b + b^{- 1}$ , the function

Γ_{b} (w) = \frac{Γ_{2} (w ∣ b, b^{- 1})}{Γ_{2} (\frac{Q}{2} ∣ b, b^{- 1})},

is invariant under $b \to b^{- 1}$ , and obeys the relations

Γ_{b} (w + b) = \sqrt{2 π} \frac{b^{b w - \frac{1}{2}}}{Γ (b w)} Γ_{b} (w), Γ_{b} (w + b^{- 1}) = \sqrt{2 π} \frac{b^{- b^{- 1} w + \frac{1}{2}}}{Γ (b^{- 1} w)} Γ_{b} (w) .

It also obeys the multiplication formula

Γ_{b} (w) = λ_{m, n, b} (m n)^{\frac{1}{4} w (b + b^{- 1} - w)} \prod_{r = 0}^{m - 1} \prod_{s = 0}^{n - 1} Γ_{b \sqrt{\frac{m}{n}}} (\frac{w + r b + s b^{- 1}}{\sqrt{m n}})

In particular, the case $m = n = 2$ is a duplication formula,

Γ_{b} (2 w) = λ_{b} 2^{w (b + b^{- 1} - 2 w)} Γ_{b} (w) Γ_{b} (w + \frac{b}{2}) Γ_{b} (w + \frac{1}{2 b}) Γ_{b} (w + \frac{b}{2} + \frac{1}{2 b})

For $ℜ w > 0$ , it has the integral representation

\log Γ_{b} (w) = \int_{0}^{\infty} \frac{d t}{t} [\frac{e^{- w t} - e^{- \frac{Q}{2} t}}{(1 - e^{- b t}) (1 - e^{- b^{- 1} t})} - \frac{{(\frac{Q}{2} - w)}^{2}}{2} e^{- t} - \frac{\frac{Q}{2} - w}{t}] .

From the function $Γ_{b} (w)$ , we define the double Sine function $S_{b} (w)$ and the Upsilon function $Υ_{b} (w)$ by

S_{b} (w) = \frac{Γ_{b} (w)}{Γ_{b} (Q - w)}, Υ_{b} (w) = \frac{1}{Γ_{b} (w) Γ_{b} (Q - w)} .

These functions obey the relations

S_{b} (w + b) = 2 \sin (π b w) S_{b} (w), Υ_{b} (w + b) = \frac{Γ (b w)}{Γ (1 - b w)} b^{1 - 2 b w} Υ_{b} (w),

plus the relations that are obtained by $b \to b^{- 1}$ . For $0 < ℜ w < ℜ Q$ they have the integral representations

\log S_{b} (w) = \int_{0}^{\infty} \frac{d t}{t} [\frac{\sinh (\frac{Q}{2} - w) t}{2 \sinh (\frac{1}{2} b t) \sinh (\frac{1}{2} b^{- 1} t)} - \frac{Q - 2 w}{t}],

\log Υ_{b} (w) = \int_{0}^{\infty} \frac{d t}{t} [{(\frac{Q}{2} - w)}^{2} e^{- t} - \frac{\sinh^{2} \frac{1}{2} (\frac{Q}{2} - w) t}{\sinh (\frac{1}{2} b t) \sinh (\frac{1}{2} b^{- 1} t)}] .

The functions $Γ_{b}, S_{b}$ and $Υ_{b}$ appear in correlation functions of two-dimensional conformal field theory, with the parameter $b$ being related to the central charge of the underlying Virasoro algebra. In particular, the three-point function of Liouville theory is written in terms of the function $Υ_{b}$ .

References

1 2 Alexanian, Shahen; Kuznetsov, Alexey (2023), "On the Barnes double gamma function", Integral Transforms and Special Functions, 34 (12): 891–914, arXiv:2208.13876, doi:10.1080/10652469.2023.2238115
1 2 Spreafico, Mauro (2009). "On the Barnes double zeta and gamma functions". Journal of Number Theory. 129 (9): 2035–2063. doi:10.1016/j.jnt.2009.03.005.
↑ De Smet, Pieter-Jan (October 23, 2026). "BarnesMultipleGamma". ResourceFunction["BarnesMultipleGamma"]. Retrieved October 24, 2026.{{cite web}}: CS1 maint: url-status (link)
1 2 Eberhardt, Lorenz (2023). "Notes on crossing transformations of Virasoro conformal blocks". Appendix B. arXiv:2309.11540 [hep-th].
↑ Ponsot, B. Recent progress on Liouville Field Theory (Thesis). arXiv:hep-th/0301193. Bibcode:2003PhDT.......180P.