TITLE: Expansion of elementary symmetric function in Jack's?
QUESTION [6 upvotes]: Consider the expansion
$$
e_\mu(x) = \sum_\lambda c_{\mu\lambda}(\alpha) J_\lambda^{(\alpha)}(x)
$$
where $J_\lambda^{(\alpha)}(x)$ are the integral-form Jack polynomials (the ones with $n!$ as coefficient of $m_{1^n}(x)$).
Is there some result which proves that each $c_{\mu\lambda}(\alpha)$ is of the form $P(\alpha)/Q(\alpha)$, where $P,Q \in \mathbb{N}[\alpha]$?
It seems that $Q$ can be chosen as some $Q_\mu(\alpha)$ that factors in a nice way (some product of deformed hook values perhaps?)

REPLY [2 votes]: Regarding my comment above, note that by induction we only need to show that $\langle J_\lambda, J_{1^n}J_\nu\rangle\in\mathbb{N}[\alpha]$. This follows from the dual Pieri formula for Jack polynomials (obtained by combining Theorems 3.3 and 6.1 of http://math.mit.edu/~rstan/pubs/pubfiles/73.pdf), so we do get $c_{\mu\lambda}(\alpha)\in\mathbb{N}[\alpha]$, as well as a factorization of $Q_\mu(\alpha)$ into linear factors.