Multiple orthogonal polynomials

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]

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

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]

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

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

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

${\displaystyle \vdots }$

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