TY - JOUR

T1 - The determinant of a hypergeometric period matrix and a generalization of Selberg's integral

AU - Richards, Donald

AU - Zheng, Qifu

N1 - Funding Information:
* Corresponding author. E-mail addresses: richards@stat.psu.edu (D. Richards), zheng@tcnj.edu (Q. Zheng). 1 This research was supported in part by the National Science Foundation grant AST-0434234. 2 This research was supported in part by a grant from the SOSA Fund of the College of New Jersey.

PY - 2007/9

Y1 - 2007/9

N2 - In an earlier paper [D. Richards, Q. Zheng, Determinant formulas for multidimensional hypergeometric period matrices, Adv. in Appl. Math. 29 (2002) 137-151] on the determinants of certain period matrices, we formulated a conjecture about the determinant of a certain hypergeometric matrix. In this article, we establish this conjecture by constructing a system of linear equations in which that determinant is one of the variables. As a consequence, we obtain the value of an integral which generalizes the well-known multidimensional beta integral of A. Selberg [A. Selberg, Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr. 26 (1944) 71-78] and some hypergeometric determinant formulas of A. Varchenko [A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 1206-1235, English translation, Math. USSR-Izv. 35 (1990) 543-571; A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990) 146-158, English translation, Math. USSR-Izv. 36 (1991) 155-167].

AB - In an earlier paper [D. Richards, Q. Zheng, Determinant formulas for multidimensional hypergeometric period matrices, Adv. in Appl. Math. 29 (2002) 137-151] on the determinants of certain period matrices, we formulated a conjecture about the determinant of a certain hypergeometric matrix. In this article, we establish this conjecture by constructing a system of linear equations in which that determinant is one of the variables. As a consequence, we obtain the value of an integral which generalizes the well-known multidimensional beta integral of A. Selberg [A. Selberg, Bemerkninger om et multipelt integral, Norsk. Mat. Tidsskr. 26 (1944) 71-78] and some hypergeometric determinant formulas of A. Varchenko [A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 1206-1235, English translation, Math. USSR-Izv. 35 (1990) 543-571; A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990) 146-158, English translation, Math. USSR-Izv. 36 (1991) 155-167].

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U2 - 10.1016/j.aam.2006.07.001

DO - 10.1016/j.aam.2006.07.001

M3 - Article

AN - SCOPUS:34547873558

SN - 0196-8858

VL - 39

SP - 395

EP - 408

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

IS - 3

ER -