Hypergeometric ladders and spherical polynomials for root systems \(BC\) of rank 2
\textit{T. Koornwinder} and \textit{I. Sprinkhuizen-Kuyper} have studied a family of generalized Jacobi polynomials in two variables, denoted by \(R_{n,k}^{\alpha,\beta,\gamma}(\xi,\eta)\), with domain of definition \[ \Omega=\{(\xi,\eta)\in\mathbb{R}^ 2:\;\eta>0,\;1-\xi+\eta>0,\;\xi^ 2>4\eta,\;0<\xi<2\} \] and orthogonality weight \[ w_{\alpha,\beta,\gamma}(\xi,\eta):=\eta^ \alpha(1-\xi+\eta)^ \beta(\xi^ 2-4\eta) ^ \gamma,\quad\text{for } (\xi,\eta)\in\Omega, \] [see SIAM J. Math. Anal. 9, 457-483 (1978; Zbl 0395.33008)]. In the present article, Deviard and Gaveau consider these polynomials in the context of functions on \(\mathbb{R}^ 2\) periodic modulo the root lattice and invariant under the Weyl group of the root system \(BC_ 2\). This approach leads to the construction of several differential operators useful in the study of the \(R_{n,k}^{\alpha,\beta,\gamma}\) (for example, eigenfunction expansion, transformation to contiguous \(n\) or \(k\) values). The authors are able to give detailed expressions for these polynomials, using a ``hypergeometric ladder'' to increase the \(k\)-index from the known expressions for \(k=0\).
scientific article; zbMATH DE number 59863
scientific article