### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 101, 33 pages      arXiv:1801.09939      https://doi.org/10.3842/SIGMA.2018.101
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

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

Ayumu Hoshino a and Jun'ichi Shiraishi b
a) Hiroshima Institute of Technology, 2-1-1 Miyake, Hiroshima 731-5193, Japan
b) Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

Received January 31, 2018, in final form September 11, 2018; Published online September 20, 2018

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 $P^{(C_n,C_n)}_{(1^r)}(x\,|\,b;q,t)= P_{(1^r)}\big(x\,|\,b^{1/2},-b^{1/2},q^{1/2}b^{1/2},-q^{1/2}b^{1/2}\,|\,q,t\big)$. It is also shown that the $q$-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,q;q,q)$ to the Hall-Littlewood polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,t;0,t)$.

Key words: Koornwinder polynomial; Catalan number.

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