×

zbMATH — the first resource for mathematics

Euler-Mellin integrals and \(A\)-hypergeometric functions. (English) Zbl 1290.32008
Summary: We consider integrals that generalize both Mellin transforms of rational functions of the form \(1/f\) and classical Euler integrals. The domains of integration of our so-called Euler-Mellin integrals are naturally related to the coamoeba of \(f\), and the components of the complement of the closure of this coamoeba give rise to a family of these integrals. After performing an explicit meromorphic continuation of Euler-Mellin integrals, we interpret them as \(A\)-hypergeometric functions and discuss their linear independence and relation to Mellin-Barnes integrals.

MSC:
32A60 Zero sets of holomorphic functions of several complex variables
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] F. Beukers, Monodromy of A-hypergeometric systems , 2011, \arxivurl arXiv: · Zbl 1229.33023
[2] —-, Irreducibility of A-hypergeometric systems , Indag. Math. (N.S.) 21 (2011), no. 1-2, 30-39. · Zbl 1229.33023
[3] E. Cattani, C. D’Andrea, and A. Dickenstein, The \(A\)-hypergeometric system associated with a monomial curve , Duke Math. J. 99 (1999), no. 2, 179-207. · Zbl 0952.33009
[4] A. Erdélyi, Beitrag zur Theorie der konfluenten hypergeometrischen Funktionen von mehreren Veränderlichen , Sitzungsber. Akad. Wiss. Wien 146 (1937), 431-467. · Zbl 0018.39902
[5] J. Forsgård, On hypersurface coamoebas and integral representations of hypergeometric functions , Licentiate thesis, Department of Mathematics, Stockholm University, 2012.
[6] J. Forsgård and P. Johansson, On the order map for hypersurface coamoebas , Ark. Mat. (to appear), \arxivurl arXiv: · Zbl 1354.32003
[7] I. Gelfand, M. Kapranov, and A. Zelevinsky, Generalized Euler integrals and \(A\)-hypergeometric functions , Adv. Math. 84 (1990), 255-271. · Zbl 0741.33011
[8] —-, Discriminants, resultants and multidimensional determinants , Birkhäuser Boston, Inc., Boston, MA, 1994.
[9] P. Johansson, Coamoebas , Licentiate thesis, Department of Mathematics, Stockholm University, 2010.
[10] L. Nilsson, Amoebas, discriminants, and hypergeometric functions , Doctoral thesis, Stockholm University, 2009.
[11] L. Nilsson and M. Passare, Discriminant coamoebas in dimension two , J. Commut. Algebra 2 (2010), no. 4, 447-471. · Zbl 1237.14062
[12] —-, Mellin transforms of multivariate rational functions , J. Geom. Anal. 23 (2013), no. 1, 24-46. · Zbl 1271.44001
[13] M. Nisse, Geometric and combinatorial structure of hypersurface coamoebas , 2009, \arxivurl arXiv:
[14] M. Nisse and F. Sottile, Non-Archimedean coamoebae , Proceedings of the 2011 Bellairs workshop in number theory: non-Archimedean and tropical geometry (O. Amini, M. Baker, X. Faber, eds.), Contemporary Mathematics, 605, American Mathematical Society, Providence, RI, 2013. · Zbl 1320.14077
[15] M. Saito, B. Sturmfels, and N. Takayama, Gröbner d eformations of hypergeometric differential equations, Springer-Verlag, Berlin, 2000.
[16] M. Schulze and U. Walther, Resonance equals reducibility for \(A\)-hypergeometric systems , Algebra Number Theory 6 (2012), no. 3, 527-537. · Zbl 1251.13023
[17] B. Sturmfels and N. Takayama, Gröbner bases and hypergeometric functions , Gröbner bases and applications (Linz, 1998), London Math. Soc. Lecture Note Ser., 251, pp. 246-258, Cambridge Univ. Press, Cambridge, 1998. · Zbl 0918.33004
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.