Multiple orthogonal polynomials for classical weights.

*(English)*Zbl 1033.33002A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. The multiple orthogonal polynomials with respect to \(p>1\) weights satisfying Pearson’s equation. In particular, a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators, is given. It is shown that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order \(p+1\). The explicit formulas and recurrence relations for these polynomials are also obtained.

Reviewer: Juri M. Rappoport (Moskva)

##### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |

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\textit{A. I. Aptekarev} et al., Trans. Am. Math. Soc. 355, No. 10, 3887--3914 (2003; Zbl 1033.33002)

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