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Sibirsk. Mat. Zh., 2012, Volume 53, Number 4, Pages 765–780 (Mi smj2362)  

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

Abstract: 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.

Keywords: 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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English version:
Siberian Mathematical Journal, 2012, 53:4, 611–624

Bibliographic databases:

UDC: 514.116.3+514.764.227+517.984+517.58
Received: 28.06.2011

Citation: 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
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\by V.~N.~Berestovski{\v\i}
\paper Zonal spherical functions on CROSS's and special functions
\jour Sibirsk. Mat. Zh.
\yr 2012
\vol 53
\issue 4
\pages 765--780
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3013525}
\transl
\jour Siberian Math. J.
\yr 2012
\vol 53
\issue 4
\pages 611--624
\crossref{https://doi.org/10.1134/S0037446612040040}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  elib
  • Сибирский математический журнал Siberian Mathematical Journal
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