Determinants of Period Matrices and an Application to Selberg's Multidimensional Beta Integral

In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990), 146–158) defined certain period matrices whose entries are Euler-type integrals representing hypergeometric functions of several variables and deri...

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Published in:Advances in applied mathematics Vol. 28; no. 3-4; pp. 602 - 633
Main Authors: Richards, Donald, Zheng, Qifu
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 01.04.2002
Elsevier
Subjects:
Binet–Cauchy formula
Exact sciences and technology
hypergeometric period matrix
Mathematical analysis
Mathematics
Measure and integration
Sciences and techniques of general use
Selberg's integral
Vandermonde determinant
hypergeometric period matrix
Binet–Cauchy formula
Selberg's integral
Vandermonde determinant
Integration
Selberg integral
Matrix algebra
Period matrix
Binet Cauchy formula
Hypergeometric function
Integral
ISSN:0196-8858, 1090-2074
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Abstract In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990), 146–158) defined certain period matrices whose entries are Euler-type integrals representing hypergeometric functions of several variables and derived remarkable closed-form expressions for the determinants of those matrices. In this article, we present elementary proofs of some of Varchenko's determinant formulas. By the same method, we obtain proofs of variations of Varchenko's determinants. As an application, we deduce new proofs of the multidimensional beta integrals of Selberg and of Aomoto. Further, we obtain a new proof of a determinant formula of A. Varchenko (Funct. Anal. Appl.25 (1999), 304–305) in which the entries are multidimensional Selberg-type integrals.
AbstractList In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990), 146–158) defined certain period matrices whose entries are Euler-type integrals representing hypergeometric functions of several variables and derived remarkable closed-form expressions for the determinants of those matrices. In this article, we present elementary proofs of some of Varchenko's determinant formulas. By the same method, we obtain proofs of variations of Varchenko's determinants. As an application, we deduce new proofs of the multidimensional beta integrals of Selberg and of Aomoto. Further, we obtain a new proof of a determinant formula of A. Varchenko (Funct. Anal. Appl.25 (1999), 304–305) in which the entries are multidimensional Selberg-type integrals.
Author Zheng, Qifu
Richards, Donald
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  organization: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey, 08540
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  givenname: Qifu
  surname: Zheng
  fullname: Zheng, Qifu
  organization: Department of Mathematics and Statistics, The College of New Jersey, Ewing, New Jersey, 08628
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Cites_doi 10.1007/BF01080091
10.3836/tjm/1270128499
10.1137/0518042
10.1007/s002220050146
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Issue 3-4
Keywords hypergeometric period matrix
Binet–Cauchy formula
Selberg's integral
Vandermonde determinant
Integration
Selberg integral
Matrix algebra
Period matrix
Binet Cauchy formula
Hypergeometric function
Integral
Language English
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Snippet In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990),...
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SubjectTerms Binet–Cauchy formula
Exact sciences and technology
hypergeometric period matrix
Mathematical analysis
Mathematics
Measure and integration
Sciences and techniques of general use
Selberg's integral
Vandermonde determinant
Title Determinants of Period Matrices and an Application to Selberg's Multidimensional Beta Integral
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