The NIST Digital Library of Mathematical Functions defines the multivariate gamma function as $$ \Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr% }\left(-\mathbf{X}\right)\left|\mathbf{X}\right|^{s_{m}-\frac{1}{2}(m+1)}\prod% _{j=1}^{m-1}|(\mathbf{X})_{j}|^{s_{j}-s_{j+1}}\mathrm{d}{\mathbf{X}}, $$ see here for all the definitions of the variables. Another representation is given by $$ \Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\pi^{m(m-1)/4}\prod_{j=1}^{m}\Gamma% \left(s_{j}-\tfrac{1}{2}(j-1)\right), $$ see here.
The book Analysis on Symmetric Cones by Faraut and Korányi (1994) has in my eyes the same definition, but in their Theorem VII.1.1. the representation suddenly has a 2 in it, which I marked in red below.
Note that to translate to the representation of
NIST Digital Library of Mathematical Functions we have $n=m(m+1)/2$, $r=m$, $d=1$. Also, see below for the definitions of all variables according to Faraut and Korányi.
Where does the 2 in Faraut and Korányi come from?
