Domains of Convergence for A-hypergeometric Series and Integrals

Nilsson, Lisa M.; Passare, Mikael; Tsikh, A. K.

doi:10.17516/1997-1397-2019-12-4-509-529

Cited by 9 publications

(7 citation statements)

References 18 publications

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“…. , q (p) ) are mutually prime) (see [8] or [9, Proposition 4.2.13]). Obviously the matrix A = (q (1) , .…”

Section: Introductionmentioning

confidence: 99%

A List of Integral Representations for the Diagonal of Power Series of a Rational Function

Senashov¹

2021

J. Sib. Fed. Univ., Math. phys.

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In this paper we present integral representations for the diagonals of power series. Such representations are obtained by lowering the multiplicity of integration for the previously known integral representation. The procedure for reducing the order of integration is carried out in the framework of the Leray theory of multidimensional residues. The concept of the amoeba of a complex analytic hypersurface plays a special role in the construction of new integral representations

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“…. , q (p) ) are mutually prime) (see [8] or [9, Proposition 4.2.13]). Obviously the matrix A = (q (1) , .…”

Section: Introductionmentioning

confidence: 99%

A List of Integral Representations for the Diagonal of Power Series of a Rational Function

Senashov¹

2021

J. Sib. Fed. Univ., Math. phys.

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“…where the vectors e i , f j and the scalars g i , h j are reals while x = (x 1 , ..., x N ) can be complex. We refer the reader to [1] for the convergence properties of MB integrals.…”

Section: Introductionmentioning

confidence: 99%

“…Progress can be made by a better understanding of N -fold MB integrals ([4] p.124), which is probably one of the reasons which motivated, in parallel to the developments in field theory mentioned above, the delivery of the first solid foundations of the difficult mathematical theory of N -fold MB integrals. The latter were presented in the rigorous approach of [8,9,10,11,12] where convergence of multiple MB integrals as well as the general properties of their series representations were studied (see also [1] and references therein). A systematic computational method of the series representations of twofold MB integrals with straight contours was also given in [10,11], allowing to handle degenerate and nondegenerate nonresonant cases 1 .…”

Section: Introductionmentioning

confidence: 99%

“…The latter were presented in the rigorous approach of [8,9,10,11,12] where convergence of multiple MB integrals as well as the general properties of their series representations were studied (see also [1] and references therein). A systematic computational method of the series representations of twofold MB integrals with straight contours was also given in [10,11], allowing to handle degenerate and nondegenerate nonresonant cases 1 . By systematic we mean that it is, for instance, possible from this method to determine the number of series representations that can be obtained for a given MB integral without performing their calculations explicitly.…”

Section: Introductionmentioning

confidence: 99%

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Multiple Series Representations of $N$-fold Mellin-Barnes Integrals

Ananthanarayan,

Banik,

Friot

et al. 2020

Preprint

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Mellin-Barnes (MB) integrals are well-known objects appearing in many branches of mathematics and physics, ranging from hypergeometric functions theory to quantum field theory, solid state physics, asymptotic theory, etc. Although MB integrals have been studied for more than one century, to our knowledge there is no systematic computational technique of the multiple series representations of N -fold MB integrals (for any given positive integer N ). We present here a breakthrough in this context, which is based on simple geometrical considerations related to the gamma functions present in the numerator of the MB integrand. The method rests on a study of N -dimensional conic hulls constructed out of normal vectors of the singular (hyper)planes associated with each of the gamma functions. Specific sets of these conic hulls are shown to be in one-to-one correspondence with series representations of the MB integral. This provides a way to express the series representations as linear combinations of multiple series, the derivation of which does not depend on the convergence properties of the latter. Our method can be applied to N -fold MB integrals with straight as well as nonstraight contours, in the resonant and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones. When convergent series are obtained the method also allows, in general, the determination of a single "master series" for each linear combination, which considerably simplifies convergence studies and numerical checks.

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“…Next in [5] (see also [6]) there was obtained a combinatorial description of the convergence domains of the series (1.4) representing solutions of the reduced equations (1.2). This description is given in terms of reciprocal positions between convergence domain and connected components of the amoeba complement for the discriminant set (see section 4).…”

Section: Introductionmentioning

confidence: 99%

Convergence of two-dimensional hypergeometric series for algebraic functions

Cherepanskiy

Tsikh

2020

Integral Transforms and Special Functions

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Description of convergence domains for multiple power series is a quite difficult problem. In 1889 J.Horn showed that the case of hypergeomteric series is more favorable. He found a parameterization formula for surfaces of conjugative radii of such series. But until recently almost nothing was known about the description of convergence domains in terms of functional inequalities ρj (|a1| , . . . , |am|) < 0 relatively moduli |ai| of series variables. In this paper we give a such description for hypergeometric series representing solutions to tetranomial algebraic equations. In our study we use the remarkable observation by M. Kapranov (1991) consisting in the fact that the Horn's formulae give a parameterization of discriminant locus for a corresponding A-discriminant. We prove that usually the considered convergence domains are determined by a signle or two inequalities ρ (|at| , |as|) ≶ 0, where ρ is a reduced discriminant.

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Domains of Convergence for A-hypergeometric Series and Integrals

Abstract: We prove two theorems on the domains of convergence for A-hypergeometric series and for associated Mellin-Barnes type integrals. The exact convergence domains are described in terms of amoebas and coamoebas of the corresponding principal A-determinants

Cited by 9 publications

References 18 publications

A List of Integral Representations for the Diagonal of Power Series of a Rational Function

A List of Integral Representations for the Diagonal of Power Series of a Rational Function

Multiple Series Representations of $N$-fold Mellin-Barnes Integrals

Convergence of two-dimensional hypergeometric series for algebraic functions

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