On perfectness of systems of complex measures satisfying Pearson's equation

Measures generating classical orthogonal polynomials (named after Hermite, Laguerre, Bessel and Jacobi) are determined by Pearson’s equation. Usually these measures are assumed to be positive, although general values of the parameters are also of interest. For general parameters, one can consider (non-Hermitian) orthogonality with respect to such complex measures supported on curves in the complex plane.

Another important generalisation is multiple orthogonality, where polynomials are assumed to satisfy orthogonality conditions with respect to several measures.

Difference Pearson's equation determines discrete measures generating Charlier, Meixner, Kravchuk and Hahn orthogonal polynomials. Here, general complex parameters may also be plugged in, which yields non-Hermitian orthogonality on discrete or continuous sets.

During this talk, we will consider systems of several same-type classical measures with complex parameters. We, in particular, going to 
find out when such systems are perfect, that is when all corresponding multiple orthogonal polynomials have proper degree and are unique up to normalisation.