Differential equation derived from generating function -- the case of disk polynomials (Q2027107)

An important characterization in classical orthogonal polynomials is one via differential equations. The main object under consideration in the paper is a partial differential equation for the disk polynomials. Given \(\nu>-1\), the sequence of polynomials \(Q^\nu_{k,j}\), \(k,j=0,1,2,\ldots\) is defined from their generating function by \[ (1-wz-\overline{wz}+w\bar{w})^{-(\nu+1)}=\sum_{k,j=0}^\infty Q^\nu_{k,j}(z)w^k \bar{w}^j, \quad |w|<1, \ |z|\le1. \] The known explicit expression for the disk polynomials enables one to derive the differential equation for such polynomials \[ 2(1-|z|^2)\,\frac{\partial^2y}{\partial\bar{z}\partial z}-(\nu+1)z\frac{\partial y}{\partial z}-(\nu+1)\bar{z}\frac{\partial y}{\partial \bar{z}} +\bigl((\nu+1)(k+j)+2kj\bigr)y=0. \] The goal of the paper is to derive the latter equation from the generating function.