I haven't found anything about Jacobi–Trudi for skew zonal polynomials.

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, $$ Z_{\lambda'}\propto\operatorname{Pf}\left((j-i)e_{r+2\ell-i-j+1}\right). $$

These results are particular cases of a theorem of Lassalle and Schlosser (Inversion of the Pieri formula for Macdonald polynomials) which gives a recurrence relation for generic zonal polynomials. This seems to be the state of the art in terms of Jacobi–Trudi-like formulas.

It is not clear to me under what circumstances these recurrence relations can be cast in the form of a determinant or a pfaffian. It is also not clear whether something similar holds for skew zonal polynomials.

I haven't found anything about Jacobi–Trudi for skew zonal polynomials.

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, $$ Z_{\lambda'}\propto\operatorname{Pf}\left((j-i)e_{r+2\ell-i-j+1}\right). $$

These results are particular cases of a theorem of Lassalle and Schlosser (Inversion of the Pieri formula for Macdonald polynomials) which gives a recurrence relation for generic zonal polynomials. This seems to be the state of the art in terms of Jacobi–Trudi-like formulas.

It is not clear to me under what circumstances these recurrence relations can be cast in the form of a determinant or a pfaffian. It is also not clear whether something similar holds for skew zonal polynomials.

I haven't found anything about Jacobi–Trudi for skew zonal polynomials.

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, $$ Z_{\lambda'}\propto{\rm Pf}\left((j-i)e_{r+2\ell-i-j+1}\right). $$

These results are particular cases of a theorem of Lassalle and Schlosser (Inversion of the Pieri formula for Macdonald polynomials) which gives a recurrence relation for generic zonal polynomials. This seems to be the state of the art in terms of Jacobi–Trudi-like formulas.

It is not clear to me under what circumstances these recurrence relations can be cast in the form of a determinant or a pfaffian. It is also not clear whether something similar holds for skew zonal polynomials.

I haven't found anything about Jacobi–Trudi for skew zonal polynomials.

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, $$ Z_{\lambda'}\propto\operatorname{Pf}\left((j-i)e_{r+2\ell-i-j+1}\right). $$

These results are particular cases of a theorem of Lassalle and Schlosser (Inversion of the Pieri formula for Macdonald polynomials) which gives a recurrence relation for generic zonal polynomials. This seems to be the state of the art in terms of Jacobi–Trudi-like formulas.

It is not clear to me under what circumstances these recurrence relations can be cast in the form of a determinant or a pfaffian. It is also not clear whether something similar holds for skew zonal polynomials.