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.^[1]

In the literature, MOPs are also called ${\displaystyle d}$-orthogonal polynomials, Hermite-Padé polynomials or polyorthogonal polynomials. MOPs should not be confused with multivariate orthogonal polynomials.

Multiple orthogonal polynomials

Consider a multiindex ${\displaystyle {\vec {n}}=(n_{1},\dots ,n_{r})\in \mathbb {N} ^{r}}$ and ${\displaystyle r}$ positive measures ${\displaystyle \mu _{1},\dots ,\mu _{r}}$ over the reals. As usual ${\displaystyle |{\vec {n}}|:=n_{1}+n_{2}+\cdots +n_{r}}$.

MOP of type 1

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

${\displaystyle \sum \limits _{j=1}^{r}\int _{\mathbb {R} }x^{k}A_{{\vec {n}},j}d\mu _{j}(x)=0,\qquad k=0,1,2,\dots ,|{\vec {n}}|-2,}$

and

${\displaystyle \sum \limits _{j=1}^{r}\int _{\mathbb {R} }x^{|{\vec {n}}|-1}A_{{\vec {n}},j}d\mu _{j}(x)=1.}$^[2]

Explanation

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

MOP of type 2

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

${\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{j}(x)=0,\qquad k=0,1,2,\dots ,n_{j}-1,\qquad j=1,\dots ,r.}$^[2]

Explanation

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

${\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{1}(x)=0,\qquad k=0,1,2,\dots ,n_{1}-1}$

${\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{2}(x)=0,\qquad k=0,1,2,\dots ,n_{2}-1}$

${\displaystyle \vdots }$

${\displaystyle \int _{\mathbb {R} }P_{\vec {n}}(x)x^{k}d\mu _{r}(x)=0,\qquad k=0,1,2,\dots ,n_{r}-1}$

Literature

• Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press. pp. 607–647. ISBN 9781107325982.
• 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