Schur polynomials $s_\lambda(x)$ have a determinantal expression. Using that, I know how to write the polynomial $s_\lambda(\frac{x}{1-x})=s_\lambda(\frac{x_1}{1-x_1},\frac{x_2}{1-x_2},...)$ as an infinite linear combination of other Schur polynomials.
Sadly, zonal polynomials do not have a determinantal expression. Still, I would like to write the zonal polynomial $Z_\lambda(\frac{x}{1-x})$ as an infinite linear combination of other zonal polynomials.
Does someone know how to do this?
Let me focus on a particular case. When $\lambda=(A,1^a)$ is a hook, the Schur function expansion is very simple. It involves only hooks and binomial coefficients: $$ s_{(A,1^a)}\left(\frac{x}{1-x}\right)=\sum_{B\ge A,b\ge a} (-1)^{a+b}{b\choose a} {B-1\choose A-1}s_{(B,1^b)}(x).$$
Maybe $ Z_{(A,1^a)}\left(\frac{x}{1-x}\right)$ has a similar expansion?
Let me report the result of some experimenting. If we write $Z_\lambda\left(\frac{x}{1-x}\right)=\sum_\mu C_{\lambda\mu}Z_\mu(x)$, then I am confident that $$ C_{(n),(m)}={m-1\choose n-1}\frac{(2n-1)!!}{(2m-1)!!}$$ and $$ C_{(1^n),(1^m)}=(-1)^{m+n}{m-1\choose n-1}\frac{(n+1)!}{(m+1)!}.$$ (I don't know how to prove these, they are conjectures). However, $C_{(3),(4,1)}=-\frac{23}{120}$ and 23 is prime, so a very simple general formula is not so likely.