# 巴尼斯G函数

${\displaystyle G(z+1)=(2\pi )^{z/2}e^{-[z(z+1)+\gamma z^{2}]/2}\prod _{n=1}^{\infty }\left[\left(1+{\frac {z}{n}}\right)^{n}e^{-z+z^{2}/(2n)}\right].}$

## 差分方程、函数方程与特殊值

${\displaystyle G(z+1)=\Gamma (z)G(z).}$

${\displaystyle G(n)={\begin{cases}0&{\mbox{if }}n=0,-1,-2,\dots \\\prod _{i=0}^{n-2}i!&{\mbox{if }}n=1,2,\dots \end{cases}}.}$

${\displaystyle G(n)={\frac {(\Gamma (n))^{n-1}}{K(n)}}.}$

${\displaystyle G(1-z)=G(1+z){\frac {1}{(2\pi )^{z}}}\exp \int _{0}^{z}\pi x\cot \pi x\,dx.}$

## 乘法公式

${\displaystyle G(nz)=K(n)n^{n^{2}z^{2}/2-nz}(2\pi )^{-{\frac {n^{2}-n}{2}}z}\prod _{i=0}^{n-1}\prod _{j=0}^{n-1}G\left(z+{\frac {i+j}{n}}\right).}$

${\displaystyle K(n)=e^{-(n^{2}-1)\zeta ^{\prime }(-1)}\cdot n^{\frac {5}{12}}\cdot (2\pi )^{(n-1)/2}\,=\,(Ae^{-{\frac {1}{12}}})^{n^{2}-1}\cdot n^{\frac {5}{12}}\cdot (2\pi )^{(n-1)/2}.}$

${\displaystyle \log \,G(z+1)}$ 渐近展开为（由巴尼斯提出）：

${\displaystyle \log G(z+1)={\frac {1}{12}}-\log A+{\frac {z}{2}}\log 2\pi +\left({\frac {z^{2}}{2}}-{\frac {1}{12}}\right)\log z-{\frac {3z^{2}}{4}}+\sum _{k=1}^{N}{\frac {B_{2k+2}}{4k\left(k+1\right)z^{2k}}}+O\left({\frac {1}{z^{2N+2}}}\right).}$

## 参考

1. ^ E.W.Barnes, "The theory of the G-function", Quarterly Journ. Pure and Appl. Math. 31 (1900), 264-314.
2. ^ M. F. Vignéras, L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL${\displaystyle (2,\mathbb {Z} )}$ , Astérisque 61, 235-249 (1979).