Representations for the parameter derivatives of some Koornwinder polynomials

In 1975, Koornwinder gave a method to construct orthogonal polynomials in two variables using the classical Jacobi polynomials. In [5], the authors introduced some new examples of Koornwinder polynomials obtained from the Koornwinder construction (see also [10]). The aim of this paper is to give the parameter derivative representations in the form of \begin{equation*} \frac{\partial P_{n,k}(\lambda;x,y)}{\partial\lambda} = \sum_{m=0}^{n-1} \sum_{j=0}^{m}d_{n,j,m}P_{m,j}(\lambda;x,y) + \sum_{j=0}^{k}e_{n,j,k}P_{n,j}(\lambda;x,y) \end{equation*} for some Koornwinder polynomials where $\lambda$ is a parameter and $0\leq k\leq n$; $n,k=0,1,2,...$ and to present orthogonality properties of the parametric derivatives of these polynomials.