Multiple orthogonal polynomials explained

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

d -orthogonal polynomials, Hermite-Padé polynomials or polyorthogonal polynomials. MOPs should not be confused with multivariate orthogonal polynomials.

Multiple orthogonal polynomials

\vec{n} = (n _{1,...,n} _{r)\in} N ^{r} and r positive measures \mu _{1,...,\mu} _{r} over the reals. As usual |\vec{n}|: = n _{1+n} _{2+ …} + n _{r} .

MOP of type 1

Polynomials

A _{\vec{n} ,j} for j =1 , 2 , ... ,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 ^{r\int} _{\R} x ^{kA} _{\vec{n} ,j}d\mu _{j(x)=0,} k =0 , 1 , 2 , ... ,|\vec{n}| -2 , and\sum\limits ^{r\int} _{\R} x ^{|\vec{n} | -1 }A _{\vec{n} ,j}d\mu _{j(x)=1.} ^[2]

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 }, ... ,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 , ... ,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

• Book: Classical and Quantum Orthogonal Polynomials in One Variable. Mourad E. H.. Ismail. Cambridge University Press. 2005. 9781107325982. 607-647.
• 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

References

1. 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
2. Book: Classical and Quantum Orthogonal Polynomials in One Variable. Mourad E. H.. Ismail. Cambridge University Press. 2005. 9781107325982. 607-608.