Multiple orthogonal polynomials

In mathematics, the multiple orthogonal polynomials (MOPs) 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 $$d$$-orthogonal polynomials, Hermite-Padé polynomials or polyorthogonal polynomials. MOPs should not be confused with multivariate orthogonal polynomials.

Multiple orthogonal polynomials
Consider a multiindex $$\vec{n}=(n_1,\dots,n_r)\in \mathbb{N}^r$$ and $$r$$ positive measures $$\mu_1,\dots,\mu_r$$ over the reals. As usual $$|\vec{n}|:=n_1+n_2+\cdots + n_r$$.

MOP of type 1
Polynomials $$A_{\vec{n},j}$$ for $$j=1,2,\dots,r$$ are of type 1 if the $$j$$-th polynomial $$A_{\vec{n},j}$$ has at most degree $$n_j-1$$ such that
 * $$\sum\limits_{j=1}^r\int_{\R}x^kA_{\vec{n},j}d\mu_j(x)=0,\qquad k=0,1,2,\dots,|\vec{n}|-2,$$

and
 * $$\sum\limits_{j=1}^r\int_{\R}x^{|\vec{n}|-1}A_{\vec{n},j}d\mu_j(x)=1.$$

Explanation
This defines a system of $$|\vec{n}|$$ equations for the $$|\vec{n}|$$ coefficients of the polynomials $$A_{\vec{n},1},A_{\vec{n},2},\dots,A_{\vec{n},r}$$.

MOP of type 2
A monic polynomial $$P_{\vec{n}}(x)$$ is of type 2 if it has degree $$|\vec{n}|$$ such that
 * $$\int_{\R}P_{\vec{n}}(x)x^k d\mu_j(x)=0,\qquad k=0,1,2,\dots,n_j-1,\qquad j=1,\dots,r.$$

Explanation
If we write $$j=1,\dots,r$$ out, we get the following definition
 * $$\int_{\R}P_{\vec{n}}(x)x^k d\mu_1(x)=0,\qquad k=0,1,2,\dots,n_1-1$$
 * $$\int_{\R}P_{\vec{n}}(x)x^k d\mu_2(x)=0,\qquad k=0,1,2,\dots,n_2-1$$
 * $$\vdots$$
 * $$\int_{\R}P_{\vec{n}}(x)x^k d\mu_r(x)=0,\qquad k=0,1,2,\dots,n_r-1$$

Literature

 * López-Lagomasino, G. (2021). An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation. In: Marcellán, F., Huertas, E.J. (eds) 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
 * López-Lagomasino, G. (2021). An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation. In: Marcellán, F., Huertas, E.J. (eds) 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