The 𝑞-Selberg polynomials for 𝑛=2

Abstract

We have conjectured that Selberg’s integral has a plethora of extensions involving the Selberg polynomials and proved that these are the Schur functions for k = 1 k = 1 . We prove this conjecture for n = 2 n = 2 and show that the polynomials are, in a formal sense, Jacobi polynomials. We conjecture an orthogonality relation for the Selberg polynomials which combines orthogonality relations for the Schur functions and Jacobi polynomials. We extend a basic Schur function identity. We give a q q -analogue of the Selberg polynomials for n = 2 n = 2 using the little q q -Jacobi polynomials.