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

Multiple orthogonal polynomials

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Consider a multiindex   and   positive measures   over the reals. As usual  .

MOP of type 1

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Polynomials   for   are of type 1 if the  -th polynomial   has at most degree   such that

 

and

 [2]

Explanation

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This defines a system of   equations for the   coefficients of the polynomials  .

MOP of type 2

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A monic polynomial   is of type 2 if it has degree   such that

 [2]

Explanation

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If we write   out, we get the following definition

 
 
 
 

Literature

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  • 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

References

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  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. ^ a b Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press. pp. 607–608. ISBN 9781107325982.