Gamma function is defined as $$ \Gamma(z) = \int\limits_{0}^{+\infty} x^{z-1} e^{-x} \; dx $$ I'm looking for multidimensional generalisation of this definition. I consider the class $Q$ of positive, concave and positively homogeneous of order one functions from $\mathbb{R}^n_+$ to $\mathbb{R}_+$. Examples of such functions are linear function $q(x) = \langle p, x \rangle$ for $p > 0$ and CES function $$ q(x) = \left( \alpha_1 x_1^{-\rho} + \ldots + \alpha_n x_n^{-\rho} \right) ^{-\frac{1}{\rho}}, \;\;\; \sum\limits_{i=1}^{n}\alpha_{i} = 1, \;\;\; \alpha_{i}, \rho > 0 \; (i = 1,\ldots, n) $$ I try to define $$ \Gamma_{q}(z) = \int\limits_{\mathbb{R}^n_+} x^{z-1}e^{-q(x)} \; dx \equiv \int\limits_{\mathbb{R}^n_+} x_1^{z_1-1}\ldots x_n^{z_n-1} e^{-q(x_1,\ldots,x_n)} \; dx_1 \ldots dx_n $$ If $n = 1$ then $\Gamma_{q}(z) = \Gamma(z)$ for any $q \in Q$.

- The main question is if there is some literature on the similar generalisation?
- If there is some other multidimensional generalisation of Gamma function?
- If there are some special types of $q(x)$ for which we can represent $\Gamma_{q}(z)$ in terms of well-known functions?