Representations for the parameter derivatives of some Koornwinder polynomials

TL;DR

This paper derives explicit parameter derivative representations for certain Koornwinder polynomials and explores their orthogonality properties, advancing understanding of their parametric behavior in multivariate orthogonal polynomial theory.

Contribution

It provides new explicit formulas for parameter derivatives of Koornwinder polynomials and investigates their orthogonality properties, extending previous work on these polynomials.

Findings

Explicit derivative representations derived for Koornwinder polynomials.

Orthogonality properties of the parametric derivatives established.

Enhanced understanding of parameter dependence in multivariate orthogonal polynomials.

Abstract

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.