In the reference
W.R. Schneider, in: S. Albeverio, G. Casati, D. Merlini (Eds.), Stochastic Processes in Classical and Quantum Systems, Lecture Notes in Physics, Vol. 262, Springer, Berlin, 1986.
the univariate Lévy-stable PDF in terms of univariate Fox H-functions is derived.
The univariate Lévy-stable PDF is defined as:
$f_{\alpha,\beta}(x)={\rm Re}\int_{0}^{\infty}dke^{-ikx}e^{-k^\alpha e^{i\frac{\pi}{2}\beta}}.$
In terms of Fox H-functions, it is expressed as:
$f_{\alpha,\beta}(x) = H^{1,1}_{2,2}\left[x \Bigg| \begin{matrix} (1-\epsilon,\epsilon),& (1-\gamma,\gamma) \\ (0,1),& (1-\gamma,\gamma) \end{matrix}\right],$
where $\epsilon=1/\alpha$ and $\gamma=\frac{\alpha-\beta}{2\alpha}$.
In the reference
S.D. Kellogg, J.W. Barnes, The bivariate H-function distribution,Mathematics and Computers in Simulation Volume 31, Issues 1–2, February 1989, Pages 91-111, https://doi.org/10.1016/0378-4754(89)90055-4
several bivariate PDFs (gamma, exponential, Gaussian-type, etc.) are expressed in terms of bivariate Fox H-functions.
I think there should exist a bivariate Lévy-stable PDF,
$f_{\alpha,\beta}(x,y)={\rm Re}\int_{0}^{\infty}dkdle^{-ikx}e^{-ily}e^{-(k^2+l^2)^{\alpha/2} e^{i\frac{\pi}{2}\beta}},$
which can be expressed in terms of bivariate Fox H-functions. Such a bivariate Lévy-stable PDF approaches the bivariate Gaussian distribution in case $\alpha\rightarrow2$ and $\beta\rightarrow0$.
Can you refer to any paper where this problem is solved?