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Zeros of Laurent multiple orthogonal polynomials on the unit circle

Rostyslav Kozhan, Marcus Vaktnäs

Abstract

We investigate two distinct formulations of Laurent multiple orthogonal polynomials on the unit circle, introduced in arXiv:2410.12094 and arXiv:2601.04783 respectively. For the first formulation, we prove that all zeros lie strictly within the complex open unit disk for any Angelesco or AT system. For the second formulation, we establish normality of all indices of the form $(\bm{n};\bm{n})$, $(\bm{n}+\bm{e}_j;\bm{n})$, and $(\bm{n};\bm{n}+\bm{e}_j)$ for any Angelesco or AT system, thereby enabling the full application of the Szegő mapping and Geronimus relations from arXiv:2601.04783 in the multiple orthogonality setting.

Zeros of Laurent multiple orthogonal polynomials on the unit circle

Abstract

We investigate two distinct formulations of Laurent multiple orthogonal polynomials on the unit circle, introduced in arXiv:2410.12094 and arXiv:2601.04783 respectively. For the first formulation, we prove that all zeros lie strictly within the complex open unit disk for any Angelesco or AT system. For the second formulation, we establish normality of all indices of the form , , and for any Angelesco or AT system, thereby enabling the full application of the Szegő mapping and Geronimus relations from arXiv:2601.04783 in the multiple orthogonality setting.
Paper Structure (12 sections, 14 theorems, 47 equations)

This paper contains 12 sections, 14 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Multiple Orthogonal Polynomials on the Unit Circle
  3. Laurent Multiple Orthogonal Polynomials on the Unit Circle
  4. General Two-Point Hermite--Padé polynomials
  5. Normality
  6. $\phi$-normality
  7. Angelesco systems on the unit circle
  8. AT systems on the Unit Circle
  9. $\Phi$-normality
  10. Multiple Paraorthogonal Polynomials
  11. Zeros of Multiple Paraorthogonal Polynomials
  12. Zeros of Multiple Orthogonal Polynomials

Key Result

Proposition 2.2

The following statements are equivalent:

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4: KVMLOPUC
  • Definition 2.5
  • Theorem 2.6: KVMLOPUC
  • ...and 27 more