Lauricella integral transformation? The Lauricella hypergeometric function $F_D$ has two different integral representations I know of. First one can be found on wikipedia:

$$\scriptsize F_D^{(n)} (a,b_1,\dots,b_n,c;x_1,\dots,x_n)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_0^1 t^{a-1}(1-t)^{c-a-1}(1-x_1t)^{-b_1}\cdots(1-x_nt)^{-b_n}~dt,\qquad \operatorname*{Re}(c)>\operatorname*{Re}(a)>0$$

And second one can be found in eq. $(1.9)$ of this paper:

$$\small \frac{\Gamma(\beta_1)\cdots \Gamma(\beta_n)\Gamma(\gamma-\beta_1-\cdots-\beta_n)}{\Gamma(\gamma)}F_D(\alpha,\beta_1,\cdots,\beta_n,\gamma,x,\cdots,x_n)$$
  $$\small =\mathop{\int\cdots\int}_{z_1,\cdots, z_n\geq 0\\\\ 1-z_1-\cdots-z_n\geq 0} z_1^{\beta_1-1}\cdots z_n^{\beta_n-1}(1-z_1-\cdots -z_n)^{\gamma-\beta_1-\cdots-\beta_n-1}\times (1-x_1 z_1 -\cdots -x_n z_n)^{-\alpha} dz_1\cdots dz_n$$

Since both definitions should be equivalent, there must be a transformation one could apply to the first integral to get the second and vice versa. Unfortunately, I could not guess what transformation that is. Does anyone knowledgeable see it and can tell me how to transform one integral into the other? Thanks for any suggestion!
 A: We want to demonstrate the equivalence of the two integral representations (3.24) and (3.25) of the hypergeometric function of type $(n+1,m+1)$ in  Aomoto and Kita (of which the Lauricella function is a special case (2,m+1)).
We start with the definition of the hypergeometric series of type $(n+1,m+1)$ as given in eq. (3.5) of Aomoto and Kita:
\begin{align}
\label{Fdef}{\tag{1}}
F(\alpha_i,\beta_j,\gamma,x)=\sum_{\nu_{ij}=0}^\infty\frac{\prod_{i=1}^n(\alpha_i)_{\sum_{j=1}^{m-n-1}\nu_{ij}}\prod_{j=1}^{m-n-1}(\beta_j)_{\sum_{i=1}^n\nu_{ij}}}{(\gamma)_{\sum_{i=1}^n\sum_{j=1}^{m-n-1}\nu_{ij}}\prod_{i=1}^n\prod_{j=1}^{m-n-1}\nu_{ij}!}\prod_{i=1}^n\prod_{j=1}^{m-n-1}x_{ij}^{\nu_{ij}},
\end{align}
where the infinite sum is over all dummy indices $\nu_{ij}$ with $i=1,2,...,n$ and $j=1,2,...,m-n-1$. Here $(x)_y$ is the Pochhammer symbol.
We will also require an integral identity given in eq. (3.17-3.19) of Aomoto and Kita:
\begin{align}
\label{id1}\tag{2}
\frac{\prod_{i=1}^n(\alpha_i)_{\sum_{j=1}^{m-n-1}\nu_{ij}}}{(\gamma)_{\sum_{i=1}^n\sum_{j=1}^{m-n-1}\nu_{ij}}}&=c_1\int_{{u_1>0,..,u_n>0}\atop{1-\sum_{i=1}^nu_i>0}}\prod_{i=1}^ndu_i\,u_i^{\sum_{j=1}^{m-n-1}\nu_{ij}+\alpha_i-1}\left(1-\sum_{i=1}^nu_i\right)^{\gamma-\sum_{i=1}^n\alpha_{i}-1},\\
c_1&=\frac{\Gamma(\gamma)}{\Gamma(\gamma-\sum_{i=1}^n\alpha_i)\prod_{i=1}^n\Gamma(\alpha_i)},
\end{align}
or alternatively, simply replacing labels and names of dummy indices, we can also write the same integral identity as:
\begin{align}
\label{id2}\tag{3}
\frac{\prod_{j=1}^{m-n-1}(\beta_j)_{\sum_{i=1}^n\nu_{ij}}}{(\gamma)_{\sum_{i=1}^n\sum_{j=1}^{m-n-1}\nu_{ij}}}&=c_2\int_{{u_1>0,..,u_n>0}\atop{1-\sum_{j=1}^{m-n-1}u_j>0}}\prod_{j=1}^{m-n-1}du_j\,u_j^{\sum_{i=1}^{n}\nu_{ij}+\beta_j-1}\left(1-\sum_{j=1}^{m-n-1}u_i\right)^{\gamma-\sum_{j=1}^{m-n-1}\beta_j-1},\\
c_2&=\frac{\Gamma(\gamma)}{\Gamma(\gamma-\sum_{j=1}^{m-n-1}\beta_j)\prod_{j=1}^{m-n-1}\Gamma(\beta_j)},
\end{align}
Now we can plug $(\ref{id1})$ into $(\ref{Fdef})$ and observe:
\begin{align}
F(\alpha_i,\beta_j,\gamma,x)=&c_1\int_{{u_1>0,..,u_n>0}\atop{1-\sum_{i=1}^nu_i>0}}\prod_{i=1}^ndu_i\,u_i^{\alpha_i-1}\left(1-\sum_{i=1}^nu_i\right)^{\gamma-\sum_{i=1}^n\alpha_{i}-1}\\
&\prod_{j=1}^{m-n-1}\sum_{\nu_{ij}=0}^\infty(\beta_j)_{\sum_{i=1}^n\nu_{ij}}\prod_{i=1}^n\frac{(u_ix_{ij})^{\nu_{ij}}}{\nu_{ij}!},\\
=&c_1\int_{{u_1>0,..,u_n>0}\atop{1-\sum_{i=1}^nu_i>0}}\prod_{i=1}^ndu_i\,u_i^{\alpha_i-1}\left(1-\sum_{i=1}^nu_i\right)^{\gamma-\sum_{i=1}^n\alpha_{i}-1}\prod_{j=1}^{m-n-1}\left(1-\sum_{i=1}^nu_ix_{ij}\right)^{-\beta_j},\notag
\end{align}
where in the last step we have recognized multinomial expansions and resummed them. This gives us the first integral representation (3.24) in Aomoto and Kita.
Similarly, we can instead plug (\ref{id2}) into (\ref{Fdef}) and analogously observe:
\begin{align}
F(\alpha_i,\beta_j,\gamma,x)=&c_2\int_{{u_1>0,..,u_n>0}\atop{1-\sum_{j=1}^{m-n-1}u_j>0}}\prod_{j=1}^{m-n-1}du_j\,u_j^{\sum_{i=1}^{n}\nu_{ij}+\beta_j-1}\left(1-\sum_{j=1}^{m-n-1}u_i\right)^{\gamma-\sum_{j=1}^{m-n-1}\beta_j-1}\\
&\prod_{i=1}^{n}\sum_{\nu_{ij}=0}^\infty(\alpha_i)_{\sum_{j=1}^{m-n-1}\nu_{ij}}\prod_{j=1}^{m-n-1}\frac{(u_jx_{ij})^{\nu_{ij}}}{\nu_{ij}!},\\
=&c_2\int_{{u_1>0,..,u_n>0}\atop{1-\sum_{j=1}^{m-n-1}u_j>0}}\prod_{j=1}^{m-n-1}du_j\,u_j^{\sum_{i=1}^{n}\nu_{ij}+\beta_j-1}\left(1-\sum_{j=1}^{m-n-1}u_i\right)^{\gamma-\sum_{j=1}^{m-n-1}\beta_j-1}\\
&\prod_{i=1}^{n}\left(1-\sum_{j=1}^{m-n-1}u_ix_{ij}\right)^{-\alpha_i}.\notag
\end{align}
Here we have again recognized and resummed multinomial expansions. This gives us the dual integral representation (3.25) in Aomoto and Kita.
With this the equivalence of the two dual representations is demonstrated.
