TY - JOUR
T1 - Determinants of period matrices and an application to Selberg's multidimensional beta integral
AU - Richards, Donald
AU - Zheng, Qifu
N1 - Funding Information:
1This research was supported by a grant from the Bell Fund to the Institute for Advanced Study. 2This research was supported by a grant from the SOSA Fund of the College of New Jersey.
PY - 2002
Y1 - 2002
N2 - 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.
AB - 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.
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U2 - 10.1006/aama.2001.0798
DO - 10.1006/aama.2001.0798
M3 - Article
AN - SCOPUS:0036083769
SN - 0196-8858
VL - 28
SP - 602
EP - 633
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
IS - 3-4
ER -