Lancaster bivariate probability distributions with Poisson, negative binomial and gamma margins

The so-called Lancaster probabilities on R2 are a class of distributions satisfying an orthogonality condition involving orthogonal polynomials with respect to their marginal laws. They are characterized in the cases where the two identical margins are Gaussian, gamma (the latter are known results, but a new treatment is given), Poisson or negative binomial distributions. Some partial results are obtained in the cases of two different Poisson or negative binomial margins, and also in the case where one margin is gamma and the other margin is negative binomial.