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?) 



Expansion of elementary symmetric function in Jack's?
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.


