Zonal polynomials may be expressed in terms of power sums as $$Z_\lambda=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_\lambda(\nu)p_\nu,$$ with usual notation in which $\omega_\lambda(\nu)$ are the zonal spherical functions. Skew zonal polynomials are defined in terms of the usual scalar product of symmetric functions by $ \langle Z_\mu Z_\nu,Z_\lambda\rangle=\langle Z_\nu,Z_{\lambda\backslash\mu}\rangle$, and we could define skew zonal spherical functions by the relation $$Z_{\lambda\backslash\mu}=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_{\lambda\backslash\mu}(\nu)p_\nu.$$

My question is whether there is a combinatorial recipe for computing $\omega_{\lambda\backslash\mu}(\nu)$ [such a recipe does exist for skew characters of the symmetric group]. I haven't found this in MacDonald's book or Stanley's paper on Jack polynomials, for instance.

Zonal polynomials may be expressed in terms of power sums as $$Z_\lambda=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_\lambda(\nu)p_\nu,$$ with usual notation in which $\omega_\lambda(\nu)$ are the zonal spherical functions. Skew zonal polynomials are defined in terms of the usual scalar product of symmetric functions by $ \langle Z_\mu Z_\nu,Z_\lambda\rangle=\langle Z_\nu,Z_{\lambda\backslash\mu}\rangle$, and we could define skew zonal spherical functions by the relation $$Z_{\lambda\backslash\mu}=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_{\lambda\backslash\mu}(\nu)p_\nu.$$

My question is whether there is a combinatorial recipe for computing $\omega_{\lambda\backslash\mu}(\nu)$ [such a recipe does exist for skew characters of the symmetric group]. I haven't found this in MacDonald's book or Stanley's paper on Jack polynomials, for instance.

EDIT: The sources I mentioned do not discuss a combinatorial interpretation of non-skew functions $\omega_\lambda(\mu)$. This is briefly discussed in Harmonic Analysis on Finite Groups (by Silberstein, Scarabotti and Tolli), but without mention of the skew case.