Combinatorics of Generalized Orthogonal Polynomials of Type RII

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.