This article is cited in 1 scientific paper (total in 1 paper)
Zonal spherical functions on CROSS's and special functions
V. N. Berestovskiĭ
Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk
We find formulas for the eigenvalues of the Laplacian and the zonal spherical functions on all simply-connected CROSS's by a simple method, using the trigonometric formulas of spherical geometry, Hopf fiber bundles, and the results on the spectra of the Laplacian on the total space and on the base of a Riemannian submersion with totally geodesic fibers. We find direct relations of the so-obtained zonal spherical functions to the special functions: hypergeometric finite Gauss series, Jacobi polynomials, and orthogonal polynomials including the ultraspherical Gegenbauer polynomials whose particular cases are given by the Legendre polynomials and the Chebyshev polynomials of the first and second kinds. We point out the relations to the corresponding results by Helgason and Berger with coauthors and give brief information about the method of calculating the spectra of the Laplacian on compact simply-connected irreducible Riemannian spaces and the spectra of the Laplacian on the CROSS's obtained therefrom.
CROSS, Hopf fiber bundle, Riemannian submersion, trigonometric formulas of spherical geometry, eigenvalues of the Laplacian, zonal spherical functions, hypergeometric functions, Jacobi polynomials, weight functions, ultraspherical polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials.
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Siberian Mathematical Journal, 2012, 53:4, 611–624
V. N. Berestovskiǐ, “Zonal spherical functions on CROSS's and special functions”, Sibirsk. Mat. Zh., 53:4 (2012), 765–780; Siberian Math. J., 53:4 (2012), 611–624
Citation in format AMSBIB
\paper Zonal spherical functions on CROSS's and special functions
\jour Sibirsk. Mat. Zh.
\jour Siberian Math. J.
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V. N. Berestovskii, I. A. Zubareva, V. M. Svirkin, “The spectra of the Laplace operators on connected compact simple Lie groups of rank 3”, Siberian Adv. Math., 26:3 (2016), 153–181
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