Opdam, E. M. Root systems and hypergeometric functions. IV. (English) Zbl 0669.33008 Compos. Math. 67, No. 2, 191-209 (1988). In three preceding papers (for part III, see the preceding review) the author and G. J. Heckmann developed a theory on multivariable hypergeometric functions associated with arbitrary root systems. In this paper the author makes further investigation on the structure of this class of functions, especially on the analytic continuation. Based on this investigation the author proves the complete integrability of three classical Hamiltonian systems that are connected with root systems. Reviewer: G.Tu Cited in 6 ReviewsCited in 50 Documents MSC: 33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 17B20 Simple, semisimple, reductive (super)algebras Keywords:multivariable hypergeometric functions; complete integrability; Hamiltonian systems; root systems Citations:Zbl 0656.17006; Zbl 0656.17007; Zbl 0669.33007 PDFBibTeX XMLCite \textit{E. M. Opdam}, Compos. Math. 67, No. 2, 191--209 (1988; Zbl 0669.33008) Full Text: Numdam EuDML References: [1] F. Calogero : Solution of the one dimensional N-body problems with quadratic and/or inversely quadratic pair potentials , J. Math. Physics 12 (1971) 419-436. · Zbl 1002.70558 [2] G.J. Heckman : Root systems and hypergeometric functions II , Comp. Math. 64 (1987) 353-373. · Zbl 0656.17007 [3] Harish-Chandra : Spherical functions on a semisimple Lie group I , Amer. J. Math. 80 (1958) 241-310. · Zbl 0093.12801 [4] S. Helgason : Groups and Geometric Analysis . Academic Press (1984). · Zbl 0543.58001 [5] L. Hörmander : An Introduction to Complex Analysis in Several Variables , van Nostrand (1966). · Zbl 0138.06203 [6] G.J. Heckman and E.M. Opdam : Root systems and hypergeometric functions I , Comp. Math. 64 (1987) 329-352. · Zbl 0656.17006 [7] C. Jacobi : Problema trium corporum mutis attractionibus cubus distantiarum inverse proportionalibus recta linea se moventium , Gesammelte Werke 4, Berlin (1866). [8] T.H. Koornwinder : Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent differential operators, I-IV , Indag. Math. 36 (1974) 48-66 and 358-381. · Zbl 0267.33008 [9] I.G. Macdonald : unpublished manuscript (1985). [10] C. Marchioro : Solution of a three body scattering problem in one dimension , J. Math. Physics 11 (1970) 2193-2196. [11] J. Moser : Three integrable Hamiltonian systems connected with isospectral deformation , Adv. Math. 16 197-220 (1975). · Zbl 0303.34019 [12] E.M. Opdam : Root systems and hypergeometric functions III , Comp. Math. 67 (1988) 21-50. · Zbl 0669.33007 [13] M.A. Olshanetsky and A.M. Perelemov : Completely integrable Hamiltonian systems connected with semisimple Lie algebras , Inv. Math. 37 (1976) 93-108. · Zbl 0342.58017 [14] J. Sekiguchi : Zonal spherical functions on some symmetric spaces , Publ. R.M.S. Kyoto. Univ. 12 Suppl. (1977) 455-459. · Zbl 0383.43005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.