# On the multidimensional generalisation of Gamma function

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 $q \in Q$.

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