On Kergin interpolation in the disk (Q789622)

Let \(K_{n,m}(x,y)\) be the Kergin interpolant of degree \(n+m-1\) to the function \(x^ ny^ m\) at the \(n+m\) points \((\cos(2k\pi /(n+m)),\quad \sin \quad(2k\pi /(n+m))), 1\leq k\leq n+m\), [see \textit{P. Kergin}, J. Approximation Theory 29, 278-293 (1980; Zbl 0492.41008)], and let \[ P_{n,m}=2^{n+m-1}\left( \begin{matrix} n+m\\ m\end{matrix} \right)(x^ ny^ m-K_{n,m})+\left( \begin{matrix} n+m\\ m\end{matrix} \right)\delta_ m, \] where \(\delta_ m=1\) if \(m\equiv 0\) mod 4, \(\delta_ m=-1\) if \(m\equiv 2\) mod 4, and \(\delta_ m=0\) otherwise. The author develops a number of relationships satisfied by the quantities \(P_{n,m}\), including recursive relationships, differential relationships, and a specific expression for \(P_{n,m}\) in terms of Chebyshev polynomials.