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 $\vec{n} = (n_{1}, \dots, n_{r}) \in ℕ^{r}$ and $r$ positive measures $μ_{1}, \dots, μ_{r}$ over the reals. As usual $| \vec{n} | := n_{1} + n_{2} + \dots + n_{r}$ .

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_{j = 1}^{r} \int_{ℝ} x^{k} A_{\vec{n}, j} d μ_{j} (x) = 0, k = 0, 1, 2, \dots, | \vec{n} | - 2,

and

\sum_{j = 1}^{r} \int_{ℝ} x^{| \vec{n} | - 1} A_{\vec{n}, j} d μ_{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 μ_{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

\int_{ℝ} P_{\vec{n}} (x) x^{k} d μ_{1} (x) = 0, k = 0, 1, 2, \dots, n_{1} - 1

\int_{ℝ} P_{\vec{n}} (x) x^{k} d μ_{2} (x) = 0, k = 0, 1, 2, \dots, n_{2} - 1

⋮

\int_{ℝ} P_{\vec{n}} (x) x^{k} d μ_{r} (x) = 0, 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