  The problem of calculating the scaled limit of the joint moments of the
characteristic polynomial, and the derivative of the characteristic polynomial,
for matrices from the unitary group with Haar measure first arose in studies
relating to the Riemann zeta function in the thesis of Hughes. Subsequently,
Winn showed that these joint moments can equivalently be written as the moments
for the distribution of the trace in the Cauchy unitary ensemble, and
furthermore relate to certain hypergeometric functions based on Schur
polynomials, which enabled explicit computations. We give a
$\beta$-generalisation of these results, where now the role of the Schur
polynomials is played by the Jack polynomials. This leads to an explicit
evaluation of the scaled moments for all $\beta > 0$, subject to the constraint
that a particular parameter therein is equal to a non negative integer.
Consideration is also given to the calculation of the moments of the singular
statistic $\sum_{j=1}^N 1/x_j$ for the Jacobi $\beta$-ensemble.
