Combinatorics of orthogonal polynomials on the unit circle Combinatorics of orthogonal polynomials on the unit circle

Orthogonal polynomials on the unit circle (OPUC for short) are a family of polynomials whose orthogonality is given by integration over the unit circle in the complex plane. There are combinatorial studies on the moments of various types of orthogonal polynomials, including classical orthogonal poly...

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Veröffentlicht in:The Ramanujan journal Jg. 68; H. 3; S. 84
Hauptverfasser: Jang, Jihyeug, Song, Minho
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.11.2025
Springer Nature B.V
Schlagworte:
Combinatorial analysis
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Numbers
Orthogonality
Partial differential equations
Polynomials
33C47
Motzkin paths
Schröder paths
05A10
Orthogonal polynomials
05A19
Moments
Łukasiewicz paths
05A15
ISSN:1382-4090, 1572-9303
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Zusammenfassung:Orthogonal polynomials on the unit circle (OPUC for short) are a family of polynomials whose orthogonality is given by integration over the unit circle in the complex plane. There are combinatorial studies on the moments of various types of orthogonal polynomials, including classical orthogonal polynomials, Laurent biorthogonal polynomials, and orthogonal polynomials of type R I . In this paper, we study the moments of OPUC from a combinatorial perspective. We provide three path interpretations for them: Łukasiewicz paths, gentle Motzkin paths, and Schröder paths. Additionally, using these combinatorial interpretations, we derive explicit formulas for the generalized moments of some examples of OPUC, including the circular Jacobi polynomials and the Rogers–Szegő polynomials. Furthermore, we introduce several kinds of generalized linearization coefficients and give combinatorial interpretations for them.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-025-01223-2