We develop a group-theoretic method of generalizing the Laplace-Beltrami operators on the classical domains. In , we defined the generalized Poisson-Cauchy transforms on the classical domains. We show that the generalized Poisson-Cauchy transforms give us eigenfunctions of the generalized Laplacians defined in this paper.
"Generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains." Proc. Japan Acad. Ser. A Math. Sci. 82 (9) 167 - 172, December 2006. https://doi.org/10.3792/pjaa.82.167