### Abstract

The article deals with orthogonal polynomials on compact infinite subsets of the complex plane. Orthogonal polynomials are treated as coordinates of generalized eigenvector of a normal operator $A$. It is shown that there exists a recursion that gives the possibility to reconstruct these polynomials. This recursion arises from generalized eigenvalue problem and, actually, this means that every gene alized eigenvector of $A$ is also a generalized eigenvector of $A^*$ with the complex conjugated eigenvalue. If the subset is actually the unit circle, it is shown that the presented algorithm is a generalization of the well-known Szego recursion from OPUC theory.

**Key words:** Block three-diagonal matrix, orthogonal polynomials, generalized eigenvector,
Verblunsky coefficients, Szeg˝o recursion, moment problem.

### Full Text

### Article Information

Title | Recursion relation for orthogonal polynomials on the complex plane |

Source | Methods Funct. Anal. Topology, Vol. 14 (2008), no. 2, 108-116 |

MathSciNet |
MR2432759 |

zbMATH |
1164.42017 |

Copyright | The Author(s) 2008 (CC BY-SA) |

### Authors Information

*Yu. M. Berezansky*

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivs'ka, Kyiv, 01601, Ukraine

*I. Ya. Ivasiuk*

Kyiv National Taras Shevchenko University, Mechanics and Mathematics Faculty, Department of Mathematical Analysis, Kyiv, 01033, Ukraine

*O. A. Mokhonko*

Kyiv National Taras Shevchenko University, Mechanics and Mathematics Faculty, Department of Mathematical Analysis, Kyiv, 01033, Ukraine

### Citation Example

Yu. M. Berezansky, I. Ya. Ivasiuk, and O. A. Mokhonko, *Recursion relation for orthogonal polynomials on the complex plane*, Methods Funct. Anal. Topology **14**
(2008), no. 2, 108-116.

### BibTex

@article {MFAT448,
AUTHOR = {Berezansky, Yu. M. and Ivasiuk, I. Ya. and Mokhonko, O. A.},
TITLE = {Recursion relation for orthogonal polynomials on the complex plane},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {14},
YEAR = {2008},
NUMBER = {2},
PAGES = {108-116},
ISSN = {1029-3531},
MRNUMBER = {MR2432759},
ZBLNUMBER = {1164.42017},
URL = {http://mfat.imath.kiev.ua/article/?id=448},
}

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