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Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers
Ayumu Hoshinoa,
Jun'ichi Shiraishib 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
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
$P^{(C_n,C_n)}_{(1^r)}(x\,|\,t;0,t)$.
Keywords:
Koornwinder polynomial; Catalan number.
MSC: 33D52;
33D45 Received: January 31, 2018; in final form
September 11, 2018; Published online
September 20, 2018
Language: English
DOI:
10.3842/SIGMA.2018.101
Fulltext:
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