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Transactions of the American Mathematical Society

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Confluence of cycles
for hypergeometric functions on $Z_{2,n+1}$


Author: Yoshishige Haraoka
Journal: Trans. Amer. Math. Soc. 349 (1997), 675-712
MSC (1991): Primary 33C60, 33C65, 33C70
DOI: https://doi.org/10.1090/S0002-9947-97-01471-2
MathSciNet review: 1321577
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Abstract: The hypergeometric function of general type, which is a generalization of the classical confluent hypergeometric functions, admits an integral representation derived from a character of a linear abelian group. For the hypergeometric function on the space of $2\times (n+1)$ matrices, a basis of cycles for the integral is constructed by a limit process, which is called a process of confluence. The determinant of the period matrix is explicitly evaluated to show the independence of the cycles.


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  • [A1] Kazuhiko Aomoto, Equations aux différences linéaires et les intégrales des fonctions multiformes. I. Théorème d’existence, Proc. Japan Acad. 50 (1974), 413–415 (French). MR 0508174
    Kazuhiko Aomoto, Équations aux différences linéaires et les intégrales des fonctions multiformes. II. Évanouissement des hypercohomologies et exemples, Proc. Japan Acad. 50 (1974), 542–545 (French). MR 0508175
    Kazuhiko Aomoto, Les équations aux différences linéaires et les intégrales des fonctions multiformes, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297 (French). MR 0508176
    Kazuhiko Aomoto, Une correction et un complément à l’article: “Les équations aux différences linéaires et les intégrales des fonctions multiformes” [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297; MR 58 #22688c], J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), no. 3, 519–523 (French). MR 560011
  • [A2] Kazuhiko Aomoto, On vanishing of cohomology attached to certain many valued meromorphic functions, J. Math. Soc. Japan 27 (1975), 248–255. MR 0440065, https://doi.org/10.2969/jmsj/02720248
  • [A3] Kazuhiko Aomoto, On the structure of integrals of power product of linear functions, Sci. Papers College Gen. Ed. Univ. Tokyo 27 (1977), no. 2, 49–61. MR 0590052
  • [A4] Kazuhiko Aomoto, Configurations and invariant Gauss-Manin connections of integrals. I, Tokyo J. Math. 5 (1982), no. 2, 249–287. MR 688126, https://doi.org/10.3836/tjm/1270214894
    Kazuhiko Aomoto, Configurations and invariant Gauss-Manin connections for integrals. II, Tokyo J. Math. 6 (1983), no. 1, 1–24. MR 707836, https://doi.org/10.3836/tjm/1270214323
  • [A5] Kazuhiko Aomoto, Configurations and invariant theory of Gauss-Manin systems, Group representations and systems of differential equations (Tokyo, 1982) Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 165–179. MR 810627
  • [G] I. M. Gel′fand, General theory of hypergeometric functions, Dokl. Akad. Nauk SSSR 288 (1986), no. 1, 14–18 (Russian). MR 841131
  • [GRS] I. M. Gel′fand, V. S. Retakh, and V. V. Serganova, Generalized Airey functions, Schubert cells and Jordan groups, Dokl. Akad. Nauk SSSR 298 (1988), no. 1, 17–21 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 1, 8–12. MR 926139
  • [HK] Yoshishige Haraoka and Hironobu Kimura, Contiguity relations of generalized confluent hypergeometric functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 5, 105–110. MR 1232148
  • [IK1] K. Iwasaki and M. Kita, Exterior power structure of the twisted de Rham cohomology associated to arrangements of hyperplanes in general position, preprint, Univ. Tokyo (1994).
  • [IK2] K. Iwasaki and M. Kita, Twisted homology of the configuration spaces of $n$-points with application to the hypergeometric functions, preprint, Univ. Tokyo (1994).
  • [Km1] H. Kimura, On rational de Rham cohomology associated with the generalized confluent hypergeometric functions I, ${\mathbb {P}}^{1}$ case, preprint (1994).
  • [Km2] H. Kimura, On Wronskian determinant of confluent hypergeometric functions, preprint (1994).
  • [Km3] H. Kimura, On the twisted homology associated with the generalized confluent hypergeometric function, private communication.
  • [KHT1] Hironobu Kimura, Yoshishige Haraoka, and Kyouichi Takano, The generalized confluent hypergeometric functions, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 9, 290–295. MR 1202635
  • [KHT2] Yoshishige Haraoka and Hironobu Kimura, Contiguity relations of generalized confluent hypergeometric functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 5, 105–110. MR 1232148
  • [KHT3] H. Kimura, Y. Haraoka and K. Takano, On contiguity relations of the confluent hypergeometric systems, Proc. Japan Acad. Ser. A 70 (1994), 47-49.
  • [Kt] Michitake Kita, On hypergeometric functions in several variables. I. New integral representations of Euler type, Japan. J. Math. (N.S.) 18 (1992), no. 1, 25–74. MR 1173830
    Michitake Kita, On hypergeometric functions in several variables. II. The Wronskian of the hypergeometric functions of type (𝑛+1,𝑚+1), J. Math. Soc. Japan 45 (1993), no. 4, 645–669. MR 1239341, https://doi.org/10.2969/jmsj/04540645
  • [MT] K. Matsumoto and N. Takayama, Braid group and a confluent hypergeometric function, J. Math. Sci. Univ. Tokyo 2 (1995), 589-610.
  • [P1] F. Pham, Introduction à l’étude topologique des singularités de Landau, Mémorial des Sciences Mathématiques, Fasc. 164, Gauthier-Villars Éditeur, Paris, 1967 (French). MR 0229263
  • [P2] Frédéric Pham, Vanishing homologies and the 𝑛 variable saddlepoint method, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 319–333. MR 713258
  • [T1] Tomohide Terasoma, A product formula for period integrals, Math. Ann. 298 (1994), no. 4, 577–589. MR 1268595, https://doi.org/10.1007/BF01459752
  • [T2] Tomohide Terasoma, Exponential Kummer coverings and determinants of hypergeometric functions, Tokyo J. Math. 16 (1993), no. 2, 497–508. MR 1247668, https://doi.org/10.3836/tjm/1270128499
  • [T3] T. Terasoma, Confluent hypergeometric functions and wild ramification, preprint.
  • [V] A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1206–1235, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 543–571. MR 1039962
    A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 146–158, 222 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 155–167. MR 1044052
    A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1206–1235, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 543–571. MR 1039962
    A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 146–158, 222 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 155–167. MR 1044052

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Additional Information

Yoshishige Haraoka
Affiliation: Department of Mathematics, Faculty of General Education, Kumamoto University, Kumamoto 860, Japan
Email: haraoka@gpo.kumamoto-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-97-01471-2
Keywords: Confluent hypergeometric function, twisted homology, confluence, period matrix
Received by editor(s): July 28, 1994
Received by editor(s) in revised form: January 9, 1995
Additional Notes: Partially supported by Grant-in-Aid for Encouragement of Young Scientists (No. 05740105), the Ministry of Education, Science and Culture, Japan
Article copyright: © Copyright 1997 American Mathematical Society