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 $\overrightarrow{n}=(n_1,\dots ,n_r)\in {\mathbb{N}}^r$ and $r$ positive measures ${\mu }_1,\dots ,{\mu }_r$ over the reals. As usual $|\overrightarrow{n}|:=n_1+n_2+\cdots +n_r$.

MOP of type 1

Polynomials $A_{\overrightarrow{n},j}$ for $j=1,2,\dots ,r$ are of type 1 if the $j$-th polynomial $A_{\overrightarrow{n},j}$ has at most degree $n_j-1$ such that

$\sum _{j=1}^r\underset{ℝ}{\int }{x}^k{A}_{\overrightarrow{n},j}d{\mu }_j(x)=0,k=0,1,2,\dots ,|\overrightarrow{n}|-2,$

and

$\sum _{j=1}^r\underset{ℝ}{\int }{x}^{|\overrightarrow{n}|-1}{A}_{\overrightarrow{n},j}d{\mu }_j(x)=1.$

Explanation

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

MOP of type 2

A monic polynomial $P_{\overrightarrow{n}}(x)$ is of type 2 if it has degree $|\overrightarrow{n}|$ such that

$\underset{ℝ}{\int }{P}_{\overrightarrow{n}}(x)x^{k}d{\mu }_j(x)=0,k=0,1,2,\dots ,n_j-1,j=1,\dots ,r.$

Explanation

If we write $j=1,\dots ,r$ out, we get the following definition

$\underset{ℝ}{\int }{P}_{\overrightarrow{n}}(x)x^{k}d{\mu }_1(x)=0,k=0,1,2,\dots ,n_1-1$

$\underset{ℝ}{\int }{P}_{\overrightarrow{n}}(x)x^{k}d{\mu }_2(x)=0,k=0,1,2,\dots ,n_2-1$

$⋮$

$\underset{ℝ}{\int }{P}_{\overrightarrow{n}}(x)x^{k}d{\mu }_r(x)=0,k=0,1,2,\dots ,n_r-1$

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