Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers

Abstract

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x|a,-a,c,-c|q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^r)}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P_{(1^r)}^{(C_n,C_n)}(x|b;q,t)=P_{(1^r)}(x|b^{1/2},-b^{1/2},q^{1/2}{b}^{1/2},-q^{1/2}{b}^{1/2}|q,t)$. It is also shown that the $q$-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials $P_{(1^r)}^{(C_n,C_n)}(x|q;q,q)$ to the Hall-Littlewood polynomials $P_{(1^r)}^{(C_n,C_n)}(x|t;0,t)$.

References in corpus (4)

• Orthogonal polynomials associated with root systems
• Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications
• Some transformation formulas associated with Askey-Wilson polynomials and Lassalle's formulas for Macdonald-Koornwinder polynomials
• Tableau Formulas for One-Row Macdonald Polynomials of Types $C_n$ and $D_n$

Cited by in corpus (2)

• An Elliptic Hypergeometric Function Approach to Branching Rules
• Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions