# Is there a Jacobi–Trudi formula for skew zonal polynomials?

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.)

For usual zonal polynomials, Kerov has shown (Generalized Hall–Littlewood symmetric functions and orthogonal polynomials) that, when $$\lambda=(r,1^s)$$ is a hook, then $$Z_{\lambda'}$$ can be written as a determinant, $$Z_{\lambda'}\propto\det\left(\frac{\lambda_i+2s-i-j+2}{\lambda_i+2s-2i+2}e_{\lambda_i-i+j}\right).$$
On the other hand, Matsumoto has shown (Two parameters circular ensembles and Jacobi–Trudi type formulas for Jack functions of rectangular shapes) that, when $$\lambda=(r^\ell)$$ is of rectangular shape, then $$Z_{\lambda'}$$ can be written as a pfaffian, $$Z_{\lambda'}\propto\operatorname{Pf}\left((j-i)e_{r+2\ell-i-j+1}\right).$$