Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 19 min ago
Peer-reviewedCC BY 4.0The Electronic Journal of Combinatorics

Combinatorics of Generalized Orthogonal Polynomials of Type RII

  • Jang Soo Kim
  • Minho Song
Open at sourcePDFDOICite

Abstract

In 1995, Ismail and Masson introduced orthogonal polynomials of types RI and RII, 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 RI 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 RII. Moreover, we generalize orthogonal polynomials of type RII 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 RI and RII. We also give a combinatorial interpretation of the generalized dual coefficients.

Related

  1. Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
    Manuel Mañas
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsMathematical physicsarXiv
  2. Hypergeometric Mixed-Type Multiple Orthogonal Polynomials
    Manuel Mañas
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsMathematical physicsarXiv
  3. Bessel Multiple Orthogonal Polynomials of Mixed Type
    Manuel Mañas
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsMathematical physicsarXiv
  4. A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point
    Anier Soria Lorente, Junior Michel
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  5. The geometric combinatorics of k-shuffle polynomials
    Chloé Postel-Vinay
    arXivPreprintarXiv licenceFull textAlgebra & number theoryAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@article{kim2026combinatorics,
  title = {{Combinatorics of Generalized Orthogonal Polynomials of Type $R_II$}},
  author = {Jang Soo Kim and Minho Song},
  journal = {The Electronic Journal of Combinatorics},
  year = {2026},
  month = oct,
  doi = {10.37236/13718},
  url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p24},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Combinatorics of Generalized Orthogonal Polynomials of Type R_II
AU  - Jang Soo Kim
AU  - Minho Song
PY  - 2026
DA  - 2026/10/23
JO  - The Electronic Journal of Combinatorics
DO  - 10.37236/13718
UR  - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p24
AB  - 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.
SN  - 1077-8926
ER  -