Reproducing kernels of weighted poly-Bergman spaces on the upper half-plane. II (Q536481)

A new form of the poly-Bergman kernels is established. It is shown that the true-\(n\)-poly-Bergman kernel is given by differentiating the rational function \[ \frac{(\overline{z} + s)^{n-1}}{\overline{\zeta} + s} \left(\frac{\zeta + s}{\overline{\zeta} + s}\right)^{n-1}. \] For weighted poly-Bergman spaces, reproducing kernels are given by means of the action of certain operator group on an orthogonal basis of \(L^2({\mathbb R}^+, \operatorname{d}\!x \operatorname{d}\!y)\).