Multiple orthogonal polynomials

In mathematics, the multiple orthogonal polynomials are orthogonal polynomials in one variable that are orthogonal with respect to a finite family of measures. The polynomials are divided into two classes named type 1 and type 2.
In the literature, MOPs are also called -orthogonal polynomials, Hermite-Padé polynomials or polyorthogonal polynomials. MOPs should not be confused with multivariate orthogonal polynomials.

Consider a multiindex and positive measures over the reals. As usual.

Polynomials for are of type 1 if the -th polynomial has at most degree such that
and

This defines a system of equations for the coefficients of the polynomials.

A monic polynomial is of type 2 if it has degree such that

If we write out, we get the following definition

Literature

López-Lagomasino, G.. An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation. In: Marcellán, F., Huertas, E.J. Orthogonal Polynomials: Current Trends and Applications. SEMA SIMAI Springer Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-56190-1_9