Parameter shift in normal generalized hypergeometric systems. (English) Zbl 0768.33016

Summary: We treat the problem of shifting parameters of \({\mathcal A}\)-hypergeometric systems when their associated toric varieties are normal. In this context we define and determine the Bernstein-Sato polynomials for the natural morphisms of shifting parameters. They are not Bernstein-Sato polynomials in a usual sense. We also give some examples.


33C70 Other hypergeometric functions and integrals in several variables
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
16W50 Graded rings and modules (associative rings and algebras)
39B32 Functional equations for complex functions
Full Text: DOI


[1] K. AOMOTO, On the vanishing of cohomology attached to certain many valued meromorphic functions, J. Math. Soc. Japan 27 (1975), 248-255. · Zbl 0301.32010
[2] K. AOMOTO, Les equations aux differences lineaires et les integrates des functions multiformes, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 22 (1975), 271-297; Une correction et un complement a article ”Les equations aux differences lineaires et les integrates des functions multiformes”, ibid. 26 (1979), 519-523. · Zbl 0339.35021
[3] K. AOMOTO, On the structure of integrals of power product of linear functions, Sci. Paper College Gen. Ed. Univ. Tokyo 27 (1977), 49-61. · Zbl 0384.35045
[4] K. AOMOTO, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 3 (1987), 191-208. · Zbl 0619.32010
[5] V. I. DANILOV, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), 85-134; Englis translation, Russian Math. Surveys 33 (1978), 97-154. · Zbl 0425.14013
[6] I. M. GELFAND, General theory of hypergeometric functions, Dokl. Akad. Nauk. SSSR 28 (1986), 14-18; English translation, Soviet Math. Dokl. 33 (1987), 9-13.
[7] I. M. GELFAND, M. I. GRAEV AND A. V. ZELEVINSKY, Holonomic system of equations and serie of hypergeometric type, Dokl. Akad. Nauk. SSSR 295 (1987), 14-19; English translation, Soviet Math. Dokl. 36 (1988), 5-10.
[8] I. M. GELFAND, A. V. ZELEVINSKY AND M. M. KAPRANOV, Equations of hypergeometric type an Newton polyhedra, Dokl. Akad. Nauk. SSSR 300 (1988), 529-534; English translation, Soviet Math. Dokl. 37 (1988), 678-683.
[9] I. M. GELFAND, A. V. ZELEVINSKY AND M. M. KAPRANOV, Hypergeometric functions and tori varieties, Funktsional. Anal, i Prilozhen 23 (1989), 12-26; English translation, Funct. Anal. Appl. 23 (1989), 94-106.
[10] I. M. GELFAND, M. M. KAPRANOV AND A. V. ZELEVINSKY, Generalized Euler integrals an A-hypergeometric functions, Advances in Mathematics 84 (1990), 255-271. · Zbl 0741.33011
[11] J. HRABOWSKI, Multiple hypergeometric functions and simple Lie algebras SL and Sp, SIAM J. Math. Anal. 16 (1985), 876-886. · Zbl 0609.33010
[12] M. -N. ISHIDA, The local cohomology groups of an affine semigroup ring, in Algebraic Geometr and Commutative Algebra in honor of Masayoshi Nagata, vol. I, Kinokuniya, Tokyo (1987), 141-153. · Zbl 0687.14002
[13] E. G. KALNINS, H. L. MANOCHA AND W. MILLER, The Lie theory of two-variable hypergeometri functions, Stud. Appl. Math. 62 (1980), 143-173. · Zbl 0452.33019
[14] T. ODA, ”Convex Bodies and Algebraic Geometry-An Introduction to the Theory of Tori Varieties, ” Ergebnisse der Math. (3) 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988. · Zbl 0628.52002
[15] M. SAITO, Normality of affine toric varieties associated with Hermitian symmetric spaces, · Zbl 0889.14024
[16] M. SAITO, Contiguity relations for the Lauricella function Fc, · Zbl 0832.33008
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.