Abstract
In this talk we give a survey of our recent results on multidimensional hypergeometric functions [GZK 1,2,7], Before developing the general theory we briefly discuss main features of the classical Gauss function F(x)= 2F1 (a,b;c;x). By definition, F(x) is the solution of the hypergeometric equation
regular at x=0 and normalized by F(0)=1. Here a,b and c are complex parameters.
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References
Aomoto K. On the structure of integrals of power products of linear functions.-Sci.Papers Coll.Gen. Educ.Univ. Tokyo, 1977, V.27, no.2, 49–61.
Appell P. , Kampé de Fêriet J. Fonctions hypergéométriques et hypersphériques; polynômes d’Hermite.-Paris, Gauthier-Villars, 1926.
Bateman H., Erdelyi A. Higher transcendental functions.-vol.1, McGraw-Hill, 1953.
Bjork J.E. Rings of differential operators.-North-Holland, 1979.
Borel A. (ed.). Seminar on intersection homology.-Birk-häuser, Boston, 1984.
Gelfand I.M. General theory of hypergeometric functions.-Doklady AN SSSR, 1986, v.288, no.1, 14–18 .
Gelfand I.M. General theory of hypergeometric functions.[Sov. Math. Dokl., 1987, V.33, 9–13]
Gelfand I.M, Gelfand S.I. Generalized hypergeometric equations.-Doklady AN SSSR, 1986, v.288, no.2, 279–283
Gelfand I.M, Gelfand S.I. Generalized hypergeometric equations.[Sov. Math. Dokl., 1987, v.33, 643–646]
Gelfand I.M., Graev M.I. A duality theorem for general hypergeometric functions.-Doklady AN SSSR, 1986, v.289, no.1, 19–23 .
Gelfand I.M., Graev M.I. A duality theorem for general hypergeometric functions. [Sov. Math. Dokl., 1987, v.34, 9–13]
Gelfand I.M., Graev M.I. Hypergeometric functions associated with the Grassmannian G3,6 .-Mat.Sbornik, 1989, v.180, no.1, 3–38
Gelfand I.M., Graev M.I. Hypergeometric functions associated with the Grassmannian G3,6 . Math. USSR Sb., 1990, v.66, no.1, 1–40.
Gelfand I.M., Graev M.I., Zelevinsky A.V. Holonomic systems of equations and series of hypergeometric type.-Doklady AN SSSR, 1987, V.295, no.1, 14–19
Gelfand I.M., Graev M.I., Zelevinsky A.V. Holonomic systems of equations and series of hypergeometric type. Sov. Math. Dokl., 1988, v.36, 5–10.
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. Equations of hypergeometric type and Newton polytopes.-Doklady AN SSSR, 1988, V.300, no.3, 529–534
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. Equations of hypergeometric type and Newton polytopes.-[Sov. Math. Dokl., 1988, v.37, 678–683].
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. Hypergeometric functions and toric varieties.-Funkc.Anal. , 1989, v.23, no.2, 12–26.
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. A-discriminants and Cayley-Koszul complexes.-Doklady AN SSSR, 1989, V.307, no.6, 1307–1310
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. A-discriminants and Cayley-Koszul complexes.-[Sov. Math. Dokl., 1990, v.40, no.1, 239–243].
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. Newton polytopes of principal A-determinants.-Doklady AN SSSR, 1989, V.308, no.1, 20–23.
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. On discriminants of polynomials in several variables.-Funkc.Anal., 1990, V.24, no.1, 1–4.
Gelfand I.M., Zelevinsky A.V., Kapranov M.M. Discriminants of polynomials in several variables and triangulations of Newton polytopes.— Algebra i analiz, 1990, v.2,no.3, 1–62.
Gelfand I.M., Kapranov M.M., Zelevinsky A.V. Generalized Euler integrals and A-hypergeometric functions.-to appear in Adv.Math.
Ginsburg V.A. Characteristic varieties and vanishing cycles.-Invent. Math, 1986, v.84, 327–402.
Horn J. Ueber die Konvergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen.-Math.Ann,1889,Bd.34, S.544–600.
Kashiwara M. Systems of micro-differential equations. Birkhäuser, Boston, 1983.
Ore O. Sur la forme de fonctions hypergeometriques de plusieurs variables.-J.Math.Pures et Appl., 1930, v.9, no. 4, 311–327.
Sato M. Singular orbits of a prehomogeneous vector space and hypergeometric finctions.-to appear in Nagoya Math.Journal.
Steenrod N. Homology with local coefficients.-Ann. Math., 1945, v.44, 610–627.
Vasiliev V.A., Gelfand I.M., Zelevinsky A.V. General hypergeometric functions on complex Grassmannians.-Funkc.Anal., 1987, V.21, no.1, 23–38.
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Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V. (1991). Hypergeometric Functions, Toric Varieties and Newton Polyhedra. In: Kashiwara, M., Miwa, T. (eds) ICM-90 Satellite Conference Proceedings. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68170-0_6
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DOI: https://doi.org/10.1007/978-4-431-68170-0_6
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