Hermite–Pade approximations of generalized hypergeometric series in two variables

We give examples of well-posed problems of joint Hermite–Pade approximations of series in two variables. We find Rodrigues formulas and integral representations for solutions. We also study the limit distribution of zeros of the corresponding polynomials. Constructions are based, on the one hand, on the classical Appel polynomials orthogonal in a triangle and, on the other hand, on various ways of proving Apery's theorem about irrationality of the number $\zeta(3)$.