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The monodromy representation and twisted period relations for Appell’s hypergeometric function F4

Part of: Singularities Deformations of analytic structures

Published online by Cambridge University Press:  11 January 2016

Yoshiaki Goto  and
Keiji Matsumoto
Yoshiaki Goto
Affiliation:
Department of Mathematics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan, [email protected]
Keiji Matsumoto
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan, [email protected]
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Abstract

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We consider the system F4 (a, b, c) of differential equations annihilating Appell's hypergeometric series F4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric series F4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of F4(a, b, c) and the twisted period relations for the fundamental systems of solutions of F4.

MSC classification

Secondary: 33C65: Appell, Horn and Lauricella functions 32G20: Period matrices, variation of Hodge structure; degenerations 32S40: Monodromy; relations with differential equations and $D$-modules
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 217 , March 2015 , pp. 61 - 94
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Aomoto, K. and Kita, M., Theory of Hypergeometric Functions, with an appendix by Kohno, T., Springer Monogr. Math., Springer, Tokyo, 2011. MR 2799182. DOI 10.1007/978-4-431-53938-4.Google Scholar
[2] Appell, P. and J., Kampé de Fériet, Fonctions hypergéométriques et hypersphériques: polynomes d'Hermite, Gauthier-Villars, Paris, 1926.Google Scholar
[3] Chaundy, T. W., An integral for Appell's hypergeometric function F(4) , Ganita 5 (1954), 231–235. MR 0083597.Google Scholar
[4] Cho, K., A generalization of Kita and Noumi's vanishing theorems of cohomology groups of local system, Nagoya Math. J. 147 (1997), 63–69. MR 1475166. Google Scholar
[5] Cho, K. and Matsumoto, K., Intersection theory for twisted cohomologies and twisted Riemann's period relations, I, Nagoya Math. J. 139 (1995), 67–86. MR 1355269.Google Scholar
[6] Goto, Y., Twisted cycles and twisted period relations for Lauricella's hypergeometric function FC ,. Internat J. Math. 24 (2013), 1350094. MR 3152203. DOI 10.1142/S0129167X13500948.Google Scholar
[7] Goto, Y., Twisted period relations for Lauricella's hypergeometric function FA , to appear in Osaka J. Math., preprint, arXiv: 1310.6088v1 [math.AG].Google Scholar
[8] Goto, Y., Kaneko, J., and Matsumoto, K., Pfaffian of Appell's hypergeometric system F4 in terms of the intersection form of twisted cohomology groups, preprint, arXiv:1502.00795v2 [math.AG].Google Scholar
[9] Haraoka, Y. and Ueno, Y., Rigidity for Appell's hypergeometric series F4 , Funkcial. Ekvac. 51 (2008), 149–164. MR 2428827. DOI 10.1619/fesi.51.149.Google Scholar
[10] Kaneko, J., Monodromy group of Appell's system (F4), Tokyo J. Math. 4 (1981), 35–54. MR 0625119. DOI 10.3836/tjm/1270215739.Google Scholar
[11] Kato, M., The irreducibilities of Appell's F4 , Ryukyu Math. J. 7 (1994), 25–34. MR 1311409.Google Scholar
[12] Kita, M. and Yoshida, M., Intersection theory for twisted cycles, I, Math. Nachr. 166 (1994), 287–304. MR 1273339.CrossRefGoogle Scholar
[13] Kita, M. and Yoshida, M., Intersection theory for twisted cycles, II: Degenerate arrangements, Math. Nachr. 168 (1994), 171–190. MR 1282638. DOI 10.1002/mana.19941680111.Google Scholar
[14] Matsumoto, K., Intersection numbers for logarithmic k-forms, Osaka J. Math. 35 (1998), 873–893. MR 1659557.Google Scholar
[15] Matsumoto, K., Monodromy and Pfaffian of Lauricella's FD in terms of the intersection forms of twisted (co)homology groups, Kyushu J. Math. 67 (2013), 367–387. MR 3115210. DOI 10.2206/kyushujm.67.367.Google Scholar
[16] Matsumoto, K., Pfaffian of Lauricella's hypergeometric system FA , preprint, arXiv:1502.00334v1 [math.AG].Google Scholar
[17] Matsumoto, K. and Yoshida, M., Monodromy of Lauricella's hypergeometric FA-system, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), 551–577. MR 3235525.Google Scholar
[18] Oshima, T., Fractional Calculus of Weyl Algebra and Fuchsian Differential Equations, MSJ Mem. 28, Math. Soc. Japan, Tokyo, 2012. MR 2986408.Google Scholar
[19] Takano, K., Monodromy group of the system for Appell's F4 , Funkcial. Ekvac. 23 (1980), 97–122. MR 0586278.Google Scholar
[20] Yoshida, M., Hypergeometric Functions, My Love. Modular Interpretations of Configuration Spaces, Aspects Math. E32, Friedr. Vieweg, Braunschweig, 1997. MR 1453580. DOI 10.1007/978-3-322-90166-8.Google Scholar