Reproducing kernel for a class of weighted Bergman spaces on the   symmetrized polydisc
  A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szeg\"{o} and the Bergman kernel on the symmetrized polydisc. 
Gadadhar Misra, Subrata Shyam Roy and Genkai Zhang
math.FA math.CV