A Proof of Some <i>q</i>-Analogues of Selberg’s Integral for $k=1$

Abstract

Selberg has given an important multiple beta type integral. We conjecture that for all $k \geqq 0$, there exists a family $\{ s_{n,\boldsymbol{\lambda} }^k ({\bf t})\} $ of homogeneous symmetric polynomials with the following property. If the integrand in Selberg’s integral is multiplied by $ s_{n,\boldsymbol{\lambda} }^k ({\bf t}) $, then the integral still has a simple closed form. For all $k \geqq 0$, this family should include the elementary symmetric functions. For $k = 1$, our symmetric functions are the Schur functions. We prove some q-analogues of this and some related results.