Skew Schur polynomials are defined as $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu\nu}s_\nu$, where the Littlewood–Richardson coefficients $c^\lambda_{\mu\nu}$ satisfy $s_\mu(x)s_\nu(x)=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda(x)$.

However, it is possible to compute the skew Schur polynomials without knowing the Littlewood–Richardson coefficients, because there is a formula expressing them as a determinant, the Jacobi–Trudi formula.

LIkewise, skew zonal polynomials are defined as $Z_{\lambda/\mu}=\sum_\nu b^\lambda_{\mu\nu}Z_\nu$, where the coefficients $b^\lambda_{\mu\nu}$ satisfy $Z_\mu(x)Z_\nu(x)=\sum_\lambda b^\lambda_{\mu\nu}Z_\lambda(x)$.

Is it possible to compute the skew zonal polynomials without knowing the $b$ coefficients, using some analogue of the Jacobi–Trudi formula? I am actually only interested in the simplest quantity $Z_{\lambda/\mu}(1^N)$.

(Is there a formula for skew Macdonald functions similar to Jacobi-Trudi identity? is a similar question about Macdonald polynomials. Zonal polynomials are a very particular case for which a lot more is known.)