Opuscula Math. 35, no. 5 (2015), 567-594
http://dx.doi.org/10.7494/OpMath.2015.35.5.567

Opuscula Mathematica

# Rigidity of monodromies for Appell's hypergeometric functions

Yoshishige Haraoka
Tatsuya Kikukawa

Abstract. For monodromy representations of holonomic systems, the rigidity can be defined. We examine the rigidity of the monodromy representations for Appell's hypergeometric functions, and get the representations explicitly. The results show how the topology of the singular locus and the spectral types of the local monodromies work for the study of the rigidity.

Keywords: rigidity, monodromy, arrangement of hyperplanes.

Mathematics Subject Classification: 33C65, 57M05.

Full text (pdf)

1. P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques - Polynomes d'Hermite, Gauthier-Villars et cie, Paris, 1926.
2. W.N. Bailey, Generalized Hypergeometric Series, Stechert-Hafner, Inc., New York, 1964.
3. R. Gérard, A.H.M. Levelt, Étude d'une classe particulière de systèmes de Pfaff du type de Fuchs sur l'espace projectif complexe, J. Math. Pures Appl. 51 (1972), 189-217.
4. Y. Haraoka, Middle convolution for completely integrable systems with logarithmic singularities along hyperplane arrangements, Adv. Stud. Pure Math. 62 (2012), 109-136.
5. Y. Haraoka, T. Matsumura, Monodromy of completely integrable systems of rank 3 singular along free divisors, preprint.
6. Y. Haraoka, Y. Ueno, Rigidity for Appell's hypergeometric series $$F_4$$, Funkcial. Ekvac. 51 (2008), 149-164.
7. E.R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255-260.
8. M. Kato, A Pfaffian system of Appell's F4, Bull. College Educ. Univ. Ryukyus 33 (1988), 331-334.
9. M. Kato, Connection formulas for Appell's system $$F_4$$ and some applications, Funkcial. Ekvac. 38 (1995), 243-266.
10. N.M. Katz, Rigid Local Systems, Princeton Univ. Press, Princeton, NJ, 1996.
11. T. Kimura, Hypergeometric Functions of Two Variables, Lecture Notes, Univ. of Minnesota, 1973.
12. P. Orlik, H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992.
13. R. Randell, The fundamental group of the complement of a union of complex hyperplanes, Invent. Math. 69 (1982), 103-108.
14. G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1922.
• Yoshishige Haraoka
• Kumamoto University, Department of Mathematics, Kumamoto 860-8555, Japan
• Tatsuya Kikukawa
• Kumamoto High School, Shin-Oe 1-8, Kumamoto 862-0972, Japan
• Communicated by P.A. Cojuhari.
• Received: 2014-02-28.
• Revised: 2014-06-09.
• Accepted: 2014-11-15.
• Published online: 2015-04-27.

Cite this article as:
Yoshishige Haraoka, Tatsuya Kikukawa, Rigidity of monodromies for Appell's hypergeometric functions, Opuscula Math. 35, no. 5 (2015), 567-594, http://dx.doi.org/10.7494/OpMath.2015.35.5.567

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.