On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions

The Selberg integral has a twin (`the Dotsenko–Fateev integral’) of the following form. We replace real variables $x_k$ in the integrand $\prod \mathrm{\mid }{x}_k{\mathrm{\mid }}^{\sigma -1}\mathrm{\mid }1-x_k{\mathrm{\mid }}^{\tau -1}\prod \mathrm{\mid }{x}_k-x_l{\mathrm{\mid }}^{2\theta }$ of the Selberg integral by complex variables $z_k$, integration over a cube we replace by an integration over the whole complex space ${\mathbb{C}}^n$. According to Dotsenko, Fateev, and Aomoto, such integral is a product of Gamma functions. We define and evaluate a family of beta integrals over spaces ${\mathbb{C}}^m\times {\mathbb{C}}^{m+1}\times \cdots \times {\mathbb{C}}^n$, which for $m=n$ gives the complex twin of the Selberg integral mentioned above (with three additional integer parameters)