On the multidimensional generalisation of Gamma function

Question

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?

Answer 1

I can suggest an answer to Question 2. E W Barnes studied multiple gamma functions early last century and there has been some sporadic recent interest in those functions. His generalization seems to be rather different from what you had in mind, but you should refer to his works, and that of recent authors, to be sure. These can be found on this Wikipedia page:

http://en.wikipedia.org/wiki/Multiple_gamma_function

The relevant references are:

• Barnes, E. W. "The Theory of the Double Gamma Function", Philos. Trans. of the Royal Society of London. Series A, 196 (1901), 265–387.
• Barnes, E. W. "On the theory of the multiple gamma function", Trans. Cambridge Philos. Soc. 19 (1904), 374–425.
• Friedman, Eduardo; Ruijsenaars, Simon. "Shintani–Barnes zeta and gamma functions", Adv. in Math. 187 (2004), no. 2, 362–395.
• Ruijsenaars, S. N. M. "On Barnes' multiple zeta and gamma functions", Adv. in Math. 156 (2000), no. 1, 107–132.