Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 014, 35 pages      arXiv:2106.03421

$q$-Selberg Integrals and Koornwinder Polynomials

Jyoichi Kaneko
Department of Mathematical Sciences, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan

Received June 23, 2021, in final form February 14, 2022; Published online February 28, 2022

We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.

Key words: Koornwinder polynomials; quadratic norm formula; antisymmetrization; $q$-Selberg integral; Mehta's integral.

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