For Jack polynomials $J_\lambda^\alpha(x_1,x_2,...,x_n)$ there exists i.e. the following relation
$$\sum_\lambda J_\lambda^\alpha(x_1,x_2,...,x_n)J_{\lambda'}^{\frac{1}{\alpha}}(y_1,y_2,...,y_m)=\prod_{i=1}^n\prod_{j=1}^m(1+x_i y_j)$$
where the sum is over all positive integer partitions $\lambda$, and $\lambda'$ is the conjugate partition. (See i.e. eq. 3.12 in terms of Macdonald polynomials here, take $q=t^\alpha$ and $t\to 1$ to reduce to Jack polynomials.) The sum over $\lambda$ terminates based on the values of $n$ and $m$. For instance, using eq. (10.15) of this paper in case when $n=m=2$, it is easy to verify explicitly that (suppressing the arguments on the $J$'s)
$$J_{2,2}^\alpha J_{2,2}^{\frac{1}{\alpha}}+J_{2,1}^\alpha J_{2,1}^{\frac{1}{\alpha}}+J_{2,0}^\alpha J_{1,1}^{\frac{1}{\alpha}}+J_{1,1}^\alpha J_{2,0}^{\frac{1}{\alpha}}+J_{1,0}^\alpha J_{1,0}^{\frac{1}{\alpha}}+J_{0,0}^\alpha J_{0,0}^{\frac{1}{\alpha}}=\prod_{i=1}^2\prod_{j=1}^2(1+x_i y_j)$$
Now I wonder if similar expansion formulas exist for cases where the partitions $\lambda$ reach arbitrarily high numbers for small $n$ and $m$? For instance, leaving $n=m=2$, do you know of an expansion of a function which involves a non-vanishing contribution of $J_\lambda^\alpha(x_1,x_2)J_\gamma^{\frac{1}{\alpha}}(y_1,y_2)$ for $\lambda\supset (2,2)$ and some partition $\gamma=\gamma(\lambda)$? Thanks for any suggestion!
