Year: 2026
Type: article
Abstract: In 1995, Ismail and Masson introduced orthogonal polynomials of types $R_I$ and $R_{II}$, which are defined by specific three-term recurrence relations with additional conditions. Recently, Kim and Stanton found a combinatorial interpretation for the moments of orthogonal polynomials of type $R_I$ in the spirit of the combinatorial theory of orthogonal polynomials due to Flajolet and Viennot. In this paper, we push this combinatorial model further to orthogonal polynomials of type $R_{II}$. Moreover, we generalize orthogonal polynomials of type $R_{II}$ by relaxing some of their conditions. Under a basis condition, we then prove a master theorem that generalizes combinatorial models for generalized moments of various types of orthogonal polynomials; ordinary orthogonal polynomials, Laurent biorthogonal polynomials, and orthogonal polynomials of types $R_I$ and $R_{II}$. We also give a combinatorial interpretation of the generalized dual coefficients.
DOI: 10.37236/13718
Authors: Jang Soo Kim, Minho Song
Institution: Sungkyunkwan University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404573840