I was wondering if someone could help me with some Jack polynomial calculations. (I use the notation of I.G. Macdonald's book "Symmetric Functions and Hall Polynomials")
Those of you familiar with Jack polynomials are surely aware of the identity $\prod_{i\geq1,j\geq1} (1-x_iy_j)^{-1/\alpha}=\sum_{\lambda}P_\lambda^{(\alpha)}(x)Q_\lambda^{(\alpha)}(y)$, where $\alpha$ is the parameter that characterises the Jack polynomials and $P$ and $Q$ are the Jack polynomials and their dual basis.
What I am trying to calculate the following expansion $\prod_{i\geq1,j\geq1}(1-x_i y_j)^{2/\alpha}=\sum_\lambda P_\lambda^{(\alpha)}(x)F_\lambda(y)$, that is, I am trying to determine the polynomials $F_\lambda(y)$ in terms of Jack polynomials $P_\mu^{(\alpha)}(y)$.
Is anyone aware of any literature that might help with this. I haven't found anything in Macdonald's book yet.