The question is involving using Sagemath and its Symmetric function package. Particularly I am interested in Jack symmetric function. So to define Jack symmetric function in sagemath

~~~

Sym = SymmetricFunctions(QQ)

Zonal = Sym.jack(t=2).J()

Schur = Sym.jack(t=2).J()

s = Sym.schur()

~~~

My interest in all these functions as expansion in power symmetric basis hence

~~~

P= Sym.p() #P() will expand the symmetric function in power symmetric basis.

~~~
$P(Zonal([2,2]))$ will give output
$$p[1, 1, 1, 1] + 2*p[2, 1, 1] + 7*p[2, 2] - 8*p[3, 1] - 2*p[4] $$
Similarly
$P(Schur([2,2]))$
$$1/12*p[1, 1, 1, 1] + 1/4*p[2, 2] + (-1/3)*p[3, 1] $$

Now the problem is that $p[2,1,1]$ it's equal to $p_2 p_1^2$ but with the sage representation I can not work by replacing $p_i $ with $p_i /h$ for example. As I know the character formula for Schur funciton I wrote my own programme using ${\bf{symmetrica.charvalue}}$ and it returns for $shur([2,1,1])$

$$1/8*p_1^4/h^4 - 1/4*p_1^2*p_2/h^3 - 1/8*p_2^2/h^2 + 1/4*p_4/h $$

My question is can I do it without writing a whole code just by using the predefined Schur function that is jack symmetric function for t =1?

The second question is about the same thing for t =2 Zonal polynomials. The zonal polynomial can be written in a power symmetric basis in Sagemath, but I want it as a product of the $p_i$ not $p[Partition]$. Is there a formula that exists in literature to do that or I can just do it in sagemath using the existing Jack function?

I can rewrite the code if I have a reference for zonal polynomial in a power symmetric basis.