Formula for multivariate Joe copula According to p. 150-151 of Cherubini et al. "Copula Methods in Finance" (2004), here are some $n$-variate Archimedean copulas:
Clayton copula
The generator is $\phi(u) = u^{-\alpha}-1$ and the Clayton copula is
$$
C(u_1, u_2, \ldots, u_n)=\left( u_1^{-\alpha}+ \ldots + u_n^{-\alpha} - n+1\right)^{-1/\alpha}, \qquad 
\alpha > 0.
$$
Gumbel copula
The generator is $\phi(u)=(-\ln(u))^\alpha$ and the Gumbel copula is
$$
C(u_1, u_2, \ldots, u_n)=\exp \bigg ( -\Big[\big(-\ln(u_1) \big )^\alpha + \ldots + \big(-\ln(u_n) \big )^\alpha \Big]^{1/\alpha}\bigg ), \qquad 
\alpha > 1.
$$
Frank copula
The generator is $\phi(u) = \ln \Big ( \frac{\exp(-\alpha u)-1}{\exp(-\alpha )-1} \Big )$ and the Frank copula is
$$
C(u_1, u_2, \ldots, u_n)=-\frac{1}{\alpha} \ln \bigg [ 1 + \frac{(e^{-\alpha u_1}-1) \cdot \ldots \cdot (e^{-\alpha u_n}-1)}{(e^{-\alpha}-1)^{n-1}}  \bigg ], \qquad 
\alpha > 0.
$$
Question: Is there an equivalent expression for an $n$-variate Joe copula (if such copula exists in the first place)?

My approach:
According to Theorem 4.10 and Definition 4.14 of Cherubini et al. (2004) (p. 149-150) as cited above, $n$-variate Archimedean copulas can be generated in a simple way. Let $\varphi$ be a strict generator, with $\varphi^{-1}$ completely monotonic on $[0,\infty]$. Then an $n$-variate Archimedean copula is the function
$$
C(u_1, \dots, u_n) = \varphi^{-1}[ (\varphi(u_1) + \dots + \varphi(u_n)].
$$
Let us try to apply this to the Joe copula. According to Table 1 on p. 6 in Krouthen "Extreme joint dependencies with copulas A new approach for the structure of C-Vines" (2015), the generator of the Joe copula is
$$
\varphi(u) = -\ln \Big( 1−(1−u)^\alpha \Big), \qquad \alpha\geq 1.
$$
After some algebra I obtain the inverse generator to be
$$
\varphi^{-1}(x) = 1-[1-\exp(-x)]^{1/\alpha}.
$$
Then the $n$-variate Joe copula then should be
$$
C(u_1, \ldots, u_n) = 1 - \bigg( 1 - \exp\big[(\ln\{1-(1−u_1)^\alpha\}+\ldots+ (\ln\{1-(1−u_n)^\alpha\}\big] \bigg)^{1/\alpha}
$$
which after some algebra simplifies to
$$
C(u_1, \ldots, u_n) = 1 - \bigg( 1 - \big[1-(1−u_1)^\alpha\big] \cdot\ldots\cdot \big[1-(1−u_n)^\alpha\big] \bigg)^{1/\alpha}.
$$
This only holds for the values of $\alpha$ that make the inverse of the generator, $\varphi^{-1}(x) = 1-[1-\exp(-x)]^{1/\alpha}$, monotonic on $[0,\infty]$. The latter set of values seems to be $\alpha\in[0,\infty)$. Combined with the requirement $\alpha\geq 1$ above, we have $\alpha\in[1,\infty)$.
Does that look OK?

Update: the following anecdotal evidence suggests it is OK. I have compared 3-variate and 4-variate Joe copulas from the copula package in R with an implementation of what I have derived above. For a concrete parameter value and a couple of concrete data points, I get matching values of the copula (cumulative density).
library(copula)

u=seq(from=0.1,to=0.9,by=0.1) # hypothetical values of a probability integral transform of a 9-variate observation
alpha=2 # the copula parameter

# 3- and 4-dimensional Joe copulas from the [`copula`][2] package
joe.cop=joeCopula(param=alpha, dim=3); print(pCopula(u[1:3], joe.cop))
joe.cop=joeCopula(param=alpha, dim=4); print(pCopula(u[1:4], joe.cop))

# 3- and 4-dimensional Joe copulas based on the derivation in this post
pcop3=1-(1-(1-(1-u[1])^alpha)*(1-(1-u[2])^alpha)*(1-(1-u[3])^alpha)                   )^(1/alpha); print(pcop3)
pcop4=1-(1-(1-(1-u[1])^alpha)*(1-(1-u[2])^alpha)*(1-(1-u[3])^alpha)*(1-(1-u[4])^alpha))^(1/alpha); print(pcop4)

