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Hypergeometric functions in several complex variables
Stockholm University, Faculty of Science, Department of Mathematics.
2002 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis deals with hypergeometric functions in several complex variables and systems of partial differential equations of hypergeometric type. One of the main objects of study in the thesis is the so-called Horn system of equations: xiPi(θ)y(x) = Qi(θ)y(x), i = 1, ..., n.Here x ∈ℂn, θ = (θ1, ..., θn),θi = xi ∂/∂xi , Pi and Qi are nonzero polynomials. By definition hypergeometric functions are (multi-valued) analytic solutions to this system of equations. The main purpose of the thesis is to systematically investigate the Horn system of equations and properties of its solutions.To construct solutions to the Horn system we use one of the variants of the Laplace transform which leads to a system of linear difference equations with polynomial coefficients. Solving this system we represent a solution to the Horn system in the form of an iterated Puiseux series.We give an explicit formula for the dimension of the space of analytic solutions to the Horn system at a generic point under some assumptions on its parameters. The proof is based on the study of the module over the Weyl algebra of linear differential operators with polynomial coefficients associated with the Horn system. Combining this formula with the theorem which allows one to represent a solution to the Horn system in the form of an iterated Puiseux series, we obtain a basis in the space of analytic solutions to this system of equations.Another object of study in the thesis is the singular set of a nonconuent hypergeometric function in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We give a description of such hypersurfaces in terms of the Newton polytopes of their defining polynomials. In particular we obtain a geometric description of the zero set of the discriminant of a general algebraic equation.In the case of two variables one can say much more about singularities of nonconuent hypergeometric functions. We give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconuent double hypergeometric series. We show in particular that the Hadamard multiplication of such series corresponds to the Minkowski sum of the Newton polytopes of polynomials whose zero loci contain the singularities of the factors.

Place, publisher, year, edition, pages
Stockholm: Matematiska institutionen , 2002. , 15 p.
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:su:diva-198ISBN: 91-7265-346-9 (print)OAI: oai:DiVA.org:su-198DiVA: diva2:190819
Public defence
2002-01-16, sal 15, hus 5, Kräftriket, Stockholm, 10:00 (English)
Opponent
Supervisors
Available from: 2001-12-27 Created: 2001-12-27 Last updated: 2009-11-12Bibliographically approved
List of papers
1. On a multidimensional system of hyprgeometric differential equations
Open this publication in new window or tab >>On a multidimensional system of hyprgeometric differential equations
1998 (English)In: Siberian mathematical journal, ISSN 0037-4466, Vol. 39, no 5, 986-997 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Springer, 1998
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-23225 (URN)10.1007/BF02672921 (DOI)
Note
Part of urn:nbn:se:su:diva-198Available from: 2001-12-27 Created: 2001-12-27 Last updated: 2010-01-04Bibliographically approved
2. On the Horn system of partial differential equations and series of hypergeometric type
Open this publication in new window or tab >>On the Horn system of partial differential equations and series of hypergeometric type
1999 (English)Report (Other academic)
Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 1999
Series
Research Reports in Mathematics, ISSN 1401-5617 ; 6
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-23226 (URN)
Note
Part of urn:nbn:se:su:diva-198Available from: 2001-12-27 Created: 2001-12-27 Last updated: 2010-01-29Bibliographically approved
3. Nonconfluent hypergeometric functions in several variables and their singularities
Open this publication in new window or tab >>Nonconfluent hypergeometric functions in several variables and their singularities
2000 (English)In: Preprints of the Max-Planck-Institut für Mathematik, no 126Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-23227 (URN)
Note
Part of urn:nbn:se:su:diva-198Available from: 2001-12-27 Created: 2001-12-27 Last updated: 2010-01-04Bibliographically approved
4. The Hadamard product of hypergeometric series
Open this publication in new window or tab >>The Hadamard product of hypergeometric series
2001 In: Research Reports in Mathematics, no 2Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-23228 (URN)
Note
Part of urn:nbn:se:su:diva-198Available from: 2001-12-27 Created: 2001-12-27Bibliographically approved

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