Schur polynomials $s_\lambda(x)$ have a determinantal expression. Using that, I know how to write the polynomial $s_\lambda(\frac{x}{1-x})$ 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?