Generalized Selberg integral

Question

I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions $$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\vert\frac{n}{2}-d\vert-\frac{1}{2}} \prod_{j<l}^d \vert u_l-u_j \vert du_1 \ldots du_d,$$ where $k_1, \ldots , k_d, n,d \in \mathbb{N}.$ I know that there exists quite some literature on this topic, I could not find any result regarding the integral above though. It would be great if someone could tell me if such a result exists and point me to the corresponding paper.

Answer 1

Your integrand is not symmetric in the variables. We must maneuver around that.

Begin by defining the integrand $$f(u)=\prod_i u_i^{-1/2}(1-u_i)^a |V(u)|$$ where $V(u)$ is the Vandermonde and $a$ is the constant you want it to be.

Your integral is $\int du f(u)(u_1^{q_1}u_2^{q_2}\cdots)$, with $q_i=k_i/2$. The point is that this integral can be written as $$\int du f(u)(u_1^{q_1}u_2^{q_2}\cdots)=\frac{1}{A}\int du f(u) M_q(u),$$ where $M_q(u)$ is the monomial symmetric function which is the symmetrized version of $(u_1^{q_1}u_2^{q_2}\cdots)$ and $A$ is the number of monomials in $M_q$ (this depends on the coincidences among the $q_i$; if they are all different, then $A=d!$).

Now we have an integral with a symmetric integrand and we can use the theory of the Jack-Selberg integral. You can read about it for instance in the paper The importance of the Selberg integral, by P Forrester, S Warnaar.

It boils down to this: You must write the function $M_q(u)$ as a linear combination of zonal polynomials $Z_\lambda(u)$, because the integral $\int du f(u)Z_\lambda(u)$ is known.

When monomial symmetric functions are written in terms of Schur functions, the coefficients are inverse Kostka numbers. I don't know about the expansion in terms of zonal polynomials.