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A bilateral extension of the $ q$-Selberg integral


Authors: Masahiko Ito and Peter J. Forrester
Journal: Trans. Amer. Math. Soc. 369 (2017), 2843-2878
MSC (2010): Primary 33D15, 33D67; Secondary 39A13
DOI: https://doi.org/10.1090/tran/6851
Published electronically: October 28, 2016
MathSciNet review: 3592530
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Abstract: A multi-dimensional bilateral $ q$-series extending the $ q$-Selberg integral is studied using concepts of truncation, regularization and connection formulae. Following Aomoto's method, which involves regarding the $ q$-series as a solution of a $ q$-difference equation fixed by its asymptotic behavior, an infinite product evaluation is obtained. The $ q$-difference equation is derived applying the shifted symmetric polynomials introduced by Knop and Sahi. As a special case of the infinite product formula, Askey-Evans's $ q$-Selberg integral evaluation and its generalization by Tarasov-Varchenko and Stokman is reclaimed, and an explanation in the context of Aomoto's setting is thus provided.


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Additional Information

Masahiko Ito
Affiliation: School of Science and Technology for Future Life, Tokyo Denki University, Tokyo 120-8551, Japan
Email: mito@cck.dendai.ac.jp

Peter J. Forrester
Affiliation: Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
Email: p.forrester@ms.unimelb.edu.au

DOI: https://doi.org/10.1090/tran/6851
Keywords: Askey--Evans's Selberg integral, Aomoto's method, connection formulae, Knop--Sahi's shifted symmetric polynomials
Received by editor(s): November 7, 2014
Received by editor(s) in revised form: September 1, 2015
Published electronically: October 28, 2016
Additional Notes: This work was supported by the Australian Research Council (Grant DP110102317) and JSPS KAKENHI Grant Number 25400118.
Article copyright: © Copyright 2016 American Mathematical Society