Poly-Bergman Type Spaces on the Siegel Domain

Abstract

We introduce poly-Bergman type spaces on the Siegel domain $D_n\subset \mathbb{C}^n$, and prove that they are isomorphic to tensor products of one-dimensional spaces generated by orthogonal polynomials of two kinds: Laguerre and Hermite polynomials. The linear span of all poly-Bergman type spaces is dense in the Hilbert space $L^2(D_n,d\mu_{\lambda})$, where $d\mu_{\lambda}=(\mathrm{Im}\, z_n |z_1|^2-\cdots -|z_{n-1}|^2)^{\lambda}dx_1dy_1\cdots dx_n dy_n$ and $\lambda>-1$.