Geodesic
Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {Ayumu Hoshino and Jun'ichi Shiraishi},
     title = {Macdonald {Polynomials} of {Type} $C_n$ with {One-Column} {Diagrams} and {Deformed} {Catalan} {Numbers}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a100/}
}
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Ayumu Hoshino; Jun'ichi Shiraishi. Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a100/

[1] Allen E. A., Combinatorial interpretations of generalizations of Catalan numbers and ballot numbers, Ph.D. Thesis, Carnegie Mellon University, 2014 | MR

[2] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54, 1985, iv+55 pp. | DOI | MR

[3] Braverman A., Finkelberg M., Shiraishi J., “Macdonald polynomials, Laumon spaces and perverse coherent sheaves”, Perspectives in Representation Theory, Contemp. Math., 610, Amer. Math. Soc., Providence, RI, 2014, 23–41, arXiv: 1206.3131 | DOI | MR | Zbl

[4] Bressoud D. M., “A matrix inverse”, Proc. Amer. Math. Soc., 88 (1983), 446–448 | DOI | MR | Zbl

[5] Feigin B., Hoshino A., Noumi M., Shibahara J., Shiraishi J., “Tableau formulas for one-row Macdonald polynomials of types $C_n$ and $D_n$”, SIGMA, 11 (2015), 100, 21 pp., arXiv: 1412.8001 | DOI | MR | Zbl

[6] Fürlinger J., Hofbauer J., “$q$-Catalan numbers”, J. Combin. Theory Ser. A, 40 (1985), 248–264 | DOI | MR | Zbl

[7] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[8] Hoshino A., Noumi M., Shiraishi J., “Some transformation formulas associated with Askey–Wilson polynomials and Lassalle's formulas for Macdonald–Koornwinder polynomials”, Mosc. Math. J., 15 (2015), 293–318, arXiv: 1406.1628 | MR | Zbl

[9] Komori Y., Noumi M., Shiraishi J., “Kernel functions for difference operators of Ruijsenaars type and their applications”, SIGMA, 5 (2009), 054, 40 pp., arXiv: 0812.0279 | DOI | MR | Zbl

[10] Koornwinder T. H., “Askey–Wilson polynomials for root systems of type $BC$”, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992, 189–204 | DOI | MR | Zbl

[11] Krattenthaler C., “A new matrix inverse”, Proc. Amer. Math. Soc., 124 (1996), 47–59 | DOI | MR | Zbl

[12] Lassalle M., “Some conjectures for Macdonald polynomials of type $B$, $C$, $D$”, Sém. Lothar. Combin., 52 (2004), B52h, 24 pp., arXiv: math.CO/0503149 | MR | Zbl

[13] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995 | MR

[14] Macdonald I. G., “Orthogonal polynomials associated with root systems”, Sém. Lothar. Combin., 45 (2000), B45a, 40 pp., arXiv: math.QA/0011046 | MR | Zbl

[15] Mimachi K., “A duality of MacDonald–Koornwinder polynomials and its application to integral representations”, Duke Math. J., 107 (2001), 265–281 | DOI | MR | Zbl

[16] Noumi M., Shiraishi J., A direct approach to the bispectral problem for the Ruijsenaars–Macdonald $q$-difference operators, arXiv: 1206.5364

[17] Okounkov A., “${\rm BC}$-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials”, Transform. Groups, 3 (1998), 181–207, arXiv: q-alg/9611011 | DOI | MR | Zbl

[18] Rains E. M., “$BC_n$-symmetric Abelian functions”, Duke Math. J., 135 (2006), 99–180, arXiv: math.CO/0402113 | DOI | MR | Zbl

[19] Rains E. M., “Transformations of elliptic hypergeometric integrals”, Ann. of Math., 171 (2010), 169–243, arXiv: math.QA/0309252 | DOI | MR | Zbl

[20] Rains E. M., Warnaar S. O., “Bounded Littlewood identities”, Mem. Amer. Math. Soc. (to appear) , arXiv: 1506.02755

[21] Shapiro L. W., “A Catalan triangle”, Discrete Math., 14 (1976), 83–90 | DOI | MR | Zbl

[22] Stokman J. V., Macdonald–Koornwinder polynomials, arXiv: 1111.6112