According to Macdonald's book, when the Jack parameter $\alpha$ is $0$,then the Jack $P$-polynomial $P_\lambda(\alpha)$ is the elementary symmetric polynomial $e_{\lambda'}$ where $\lambda'$ is the dual integer partition of $\lambda$. What about the skew Jack $P$-polynomial $P_{\lambda/\mu}(\alpha)$ when $\alpha=0$? I experimented a little with my implementation of these polynomials and it seems that it is proportional to some elementary symmetric polynomial, but I have not been able to derive a precise conjecture.
EDIT
This observation is wrong:
it seems that it is proportional to some elementary symmetric polynomial
Now I observed that $P_{\lambda/\mu}(0)$ is a linear combination of some elementary symmetric polynomials $e_{\nu'}$, and the integer partitions $\nu'$ are the dual partitions of some integer partitions $\nu$ such that the Littlewood-Richardson coefficient $c^{\lambda}_{\mu,\nu}$ is not zero.
