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

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