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Selberg integral theory and Muttalib--Borodin ensembles

We study Muttalib--Borodin ensembles --- particular eigenvalue PDFs on the half-line --- with classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We show how the theory of the Selberg integral, involving also Jack and Schur polynomials, naturally leads to a multi-parameter generalisation of these particular Muttalib--Borodin ensembles, and also to the explicit form of underlying biorthogonal polynomials of a single variable. A suitable generalisation of the original definition of the Muttalib--Borodin ensemble allows for negative eigenvalues. In the cases of generalised Gaussian, symmetric Jacobi and Cauchy weights, we show that the problem of computing the normalisations and the biorthogonal polynomials can be reduced down to Muttalib--Borodin ensembles with classical weights on the positive half-line.