ScholarWorks@성균관대학교

초록

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 RI. In this paper, we study the moments of OPUC from a combinatorial perspective. We provide three path interpretations for them: Lukasiewicz paths, gentle Motzkin paths, and Schr & ouml;der 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-Szego polynomials. Furthermore, we introduce several kinds of generalized linearization coefficients and give combinatorial interpretations for them.

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