Classical skew orthogonal polynomials in a two-component log-gas with charges $+1$ and $+2$

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.