# The coefficients of the Jack polynomials are polynomials in the Jack parameter

I implemented the Jack polynomials with a symbolic Jack parameter $$\alpha$$ in their coefficients ($$\alpha=1$$ for Schur polynomials, $$\alpha=2$$ for zonal polynomials). From my implementation (and also from the definition I think), the coefficients are fractions of polynomials in $$\alpha$$. But each time I compute a $$J$$-Jack polynomial, I observe that they actually are polynomials in $$\alpha$$ (I didn't check yet for the $$P$$-Jack polynomials and the $$Q$$-Jack polynomials). Is it an established fact, and are there some references about it?

• Coefficients in what basis? Apr 2 at 12:59
• the combinatorial formula in Wikipedia is a polynomial in $\alpha$: en.wikipedia.org/wiki/Jack_function Apr 2 at 13:03
• But being a symmetric function, you can group together all the terms for the different permutations of a monomial into the monomial symmetric function, no? (Anyways sorry this discussion is probably irrelevant seeing as you got the answer you were looking for...) Apr 2 at 13:26
• @SamHopkins Thank you for this remark! This can considerably shorten the expressions of the Jack polynomials. I never had this idea. Apr 2 at 14:56
• @CarloBeenakker, re, since that seems to answer the question, maybe you can post it as an answer? Apr 2 at 15:04

the combinatorial formula in Wikipedia, due to Knop and Sahi, is a polynomial in $$\alpha$$.
• And to settle a subquestion: by definition, the '$P$-normalisation' is monic, $P_\lambda = m_\lambda + \text{lower}$, so the lower terms have coefficients that are rational functions of $\alpha$. It differs from the '$J$-normalisation by a factor that depends on $\alpha$ and remove the denominator. Apr 7 at 14:04