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.



Generalized Selberg integral
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](https://www.ams.org/journals/bull/2008-45-04/S0273-0979-08-01221-4/) *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.


