  There is a two-component log-gas system with Boltzmann factor which provides
an interpolation between the eigenvalue PDF for $\beta = 1$ and $\beta = 4$
invariant random matrix ensembles. The solvability of this log-gas system
relies on the construction of particular skew orthogonal polynomials, with the
skew inner product a linear combination of the $\beta = 1$ and $\beta = 4$
inner products, each involving weight functions. For suitably related classical
weight functions, we seek to express the skew orthogonal polynomials as linear
combinations of the underlying orthogonal polynomials. It is found that in each
case (Gaussian, Laguerre, Jacobi and generalised Cauchy) the coefficients can
be expressed in terms of hypergeometric polynomials with argument relating to
the fugacity. In the Jacobi case, for example, these are a special case of the
Wilson polynomials.
