# zbMATH — the first resource for mathematics

Monodromy of $$A$$-hypergeometric functions. (English) Zbl 1355.33017
Author’s abstract: Using Mellin-Barnes integrals we give a method to compute elements of the monodromy group of an A-hypergeometric system of differential equations. The method works under the assumption that the A-hypergeometric system has a basis of solutions consisting of Mellin-Barnes integrals. Hopefully these elements generate the full monodromy group , but this has only been verified in some special cases.

##### MSC:
 33C70 Other hypergeometric functions and integrals in several variables 33C65 Appell, Horn and Lauricella functions 34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
Full Text:
##### References:
 [1] Adolphson A., Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), 269-290. · Zbl 0804.33013 [2] Andrews G. E., Askey R. and Roy R., Special functions, Encyclopedia Math. Appl. 71, Cambridge University Press, Cambridge 1999. [3] Antipova I. A., Inversion of multidimensional Mellin transforms, Russian Math. Surveys 62 (2007), 977-979. · Zbl 1148.44003 [4] Antipova I. A., Inversion of many-dimensional Mellin transforms and solutions of algebraic equations, Sb. Math. 198 (2007), 474-463. · Zbl 1142.44006 [5] Beukers F., Algebraic A-hypergeometric functions, Invent. Math. 180 (2010), 589-610. · Zbl 1251.33011 [6] Beukers F., Irreducibility of A-hypergeometric systems, Indag. Math. (N.S.) 21 (2011), 30-39. · Zbl 1229.33023 [7] Beukers F., Notes on A-hypergeometric functions, Arithmetic and Galois theories of differential equations, Sémin. Congr. 23, Société Mathématique de France, Paris (2011), 25-61. · Zbl 1295.33015 [8] Beukers F. and Heckman G., Monodromy for the hypergeometric function $${{}_{n}F_{n-1}}$$, Invent. Math. 95 (1989), 325-354. · Zbl 0663.30044 [9] Chen Y.-H., Yang Y. and Yui N., Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds, J. reine angew. Math. 616 (2008), 167-203. · Zbl 1153.34055 [10] Deligne P. and Mostow G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 5-89. · Zbl 0615.22008 [11] Gelfand I. M., M. I.Graev , Zelevinsky A. V., Holonomic systems of equations and series of hypergeometric type (in Russian), Dokl. Akad. Nauk SSSR 295 (1987), 14-19. [12] Gelfand I. M., Kapranov M. M., Zelevinsky A. V., Generalized Euler integrals and A-hypergeometric functions, Adv. Math 84 (1990), 255-271. · Zbl 0741.33011 [13] Gelfand I. M., Kapranov M. M. and Zelevinsky A. V., A correction to the paper “Hypergeometric equations and toral manifolds”, Funct. Anal. Appl. 27 (1993), 295-295. [14] Gelfand I. M., Zelevinsky A. V. and Kapranov M. M., Equations of hypergeometric type and Newton polytopes (in Russian), Dokl. Akad. Nauk SSSR 300 (1988), 529-534. [15] Gelfand I. M., Zelevinsky A. V. and Kapranov M. M., Hypergeometric functions and toral manifolds, Funct. Anal. Appl. 23 (1989), 94-106. · Zbl 0721.33006 [16] Goto Y., The monodromy representation of Lauricella’s hypergeometric function $${F_{C}}$$, preprint 2014, . · Zbl 1362.33017 [17] Hanamura M. and Yoshida M., Hodge structure on twisted cohomologies and twisted Riemann inequalities I, Nagoya Math. J. 154 (1999), 123-139. · Zbl 0956.14020 [18] Haraoka Y. and Ueno Y., Rigidity for Appell’s hypergeometric series $${F_{4}}$$, Funkcial. Ekvac. 51 (2008), 149-164. · Zbl 1167.33006 [19] Kaneko J., Monodromy group of Appell’s system $${F_{4}}$$, Tokyo J. Math 4 (1981), 35-54. · Zbl 0474.33010 [20] Kato M., Appell’s hypergeometric systems $${F_{2}}$$ with finite irreducible monodromy groups, Kyushu J. Math. 54 (2000), 279-305. · Zbl 0970.33009 [21] Kita M. and Yoshida M., Intersection theory for twisted cycles, Math. Nachr. 166 (1994), 287-304; Intersection theory for twisted cycles II, Math. Nachr. 168 (1994), 171-190. · Zbl 0847.32043 [22] Maclachlan N. W., Complex variable theory and transform calculus, 2nd ed., Cambridge University Press, Cambridge 1953. [23] Matsumoto K., Sasaki T., Takayama N. and Yoshida M., Monodromy of the hypergeometric equation of type $${(3,6)}$$. I, Duke Math. J. 71 (1993), 403-426. · Zbl 0799.33008 [24] Matsumoto K., Sasaki T., Takayama N. and Yoshida M., Monodromy of the hypergeometric equation of type $${(3,6)}$$. II: The unitary reflection group of order $${2^{9}· 3^{7}· 5· 7}$$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 20 (1993), 617-631. · Zbl 0810.33009 [25] Matsumoto K. and Yoshida M., Monodromy of Lauricella’s hypergeometric $${F_{A}}$$-system, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), 551-577. · Zbl 1314.32045 [26] Mimachi K., Intersection numbers for twisted cycles and the connection problem associated with the generalized hypergeometric function $${{}_{n+1}F_{n}}$$, Int. Math. Res. Not. IMRN 2011 (2011), 1757-1781. · Zbl 1216.33014 [27] Nilsson L., Amoebas, discriminants, and hypergeometric functions, PhD dissertation, Stockholm University 2009. [28] Nørlund N. E., Hypergeometric functions, Acta Math. 94 (1955), 289-349. · Zbl 0067.29402 [29] Picard E., Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques, Ann. Éc. Norm. Supér. (2) 10 (1881), 304-322. · JFM 13.0389.01 [30] Saito M., Sturmfels B. and Takayama N., Gröbner deformations of hypergeometric differential equations, Algorithms Comput. Math. 6, Springer-Verlag, Berlin 2000. [31] Sasaki T., On the finiteness of the monodromy group of the system of hypergeometric differential equations $${(F_{D})}$$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 565-573. · Zbl 0388.33003 [32] Schulze M. and Walther U., Resonance equals reducibility for A-hypergeometric systems, Algebra Number Theory 6 (2012), 527-537. · Zbl 1251.13023 [33] Smith F. C., Relations among the fundamental solutions of the generalized hypergeometric equation when $${p=q+1}$$. Non-logarithmic cases, Bull. Amer. Math. Soc. 44 (1938), 429-433. · JFM 64.0334.02 [34] Stienstra J., GKZ hypergeometric structures, Arithmetic and geometry around hypergeometric functions, Progr. Math. 260, Birkhäuser-Verlag, Basel (2007), 313-371. · Zbl 1119.14003 [35] Takano K., Monodromy of the system for Appell’s $${F_{4}}$$, Funkcial. Ekvac. 23 (1980), 97-122. · Zbl 0441.33009 [36] Terada T., Fonctions hypergéométriques F1 et fonctions automorphes I, J. Math. Soc. Japan 35 (1983), 451-475. · Zbl 0506.33001 [37] Yoshida M., Hypergeometric functions, my love. Modular interpretations of configuration spaces, Aspects Math. 32, Vieweg-Verlag, Wiesbaden 1997. · Zbl 0889.33008 [38] Zhdanov O. N. and Tsikh A. K., Studying the multiple Mellin-Barnes integrals by means of multidimensional residues, Sib. Math. J. 39 (1998), 245-260. · Zbl 0944.32003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.