Quelques valeurs prises par les polynômes de Macdonald décalés
Annales de l'Institut Fourier, Volume 49 (1999) no. 2, p. 543-561
We give explicit expressions for some generalized binomial coefficients associated with Macdonald polynomials, i.e., the value at some special points of shifted Macdonald polynomials. These expressions are obtained in terms of q-basic hypergeometric series.
Nous explicitons la valeur de certains des coefficients binomiaux généralisés associés aux polynômes de Macdonald, c’est-à-dire la valeur en certains points particuliers des polynômes de Macdonald décalés. Ces expressions font intervenir les fonctions hypergéométriques de base q.
@article{AIF_1999__49_2_543_0,
     author = {Lassalle, Michel},
     title = {Quelques valeurs prises par les polyn\^omes de Macdonald d\'ecal\'es},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {2},
     year = {1999},
     pages = {543-561},
     doi = {10.5802/aif.1682},
     zbl = {0930.11006},
     mrnumber = {1697372},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1999__49_2_543_0}
}
Lassalle, Michel. Quelques valeurs prises par les polynômes de Macdonald décalés. Annales de l'Institut Fourier, Volume 49 (1999) no. 2, pp. 543-561. doi : 10.5802/aif.1682. http://www.numdam.org/item/AIF_1999__49_2_543_0/

[1] J. Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal., 24 (1993), 1086-1110. | Zbl 0783.33008

[2] F. Knop, Symmetric and non-symmetric quantum Capelli polynomials, to appear. | Zbl 0954.05049

[3] M. Lassalle, Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal., 158 (1998), 289-324. | Zbl 0914.33012

[4] M. Lassalle, Une formule du binôme généralisée pour les polynômes de Jack, C. R. Acad. Sci. Paris, Sér. I Math., 310 (1990), 253-256. | Zbl 0698.33010

[5] M. Lassalle, Coefficients du binôme généralisés, C. R. Acad. Sci. Paris, Sér. I Math., 310 (1990), 257-260. | Zbl 0733.33001

[6] M. Lassalle, Some combinatorial conjectures for shifted Jack polynomials, Ann. Combinatorics, 2 (1998), 145-163. | Zbl 0933.33014

[7] I. G. Macdonald, Symmetric functions and Hall polynomials, second edition, Clarendon Press, Oxford, 1995. | Zbl 0824.05059

[8] A. Okounkov, (Shifted) Macdonald polynomials: q-integral representation and combinatorial formula, Compositio Math., 112 (1998), 147-182. | Zbl 0897.05085

[9] A. Okounkov, Binomial formula for Macdonald polynomials and applications, Math. Res. Lett., 4 (1997), 533-553. | Zbl 0911.33012

[10] A. Okounkov, G. Olshanski, Shifted Jack polynomials, binomial formula and applications, Math. Res. Lett., 4 (1997), 69-78. | Zbl 01019787

[11] A. Okounkov, G. Olshanski, Shifted Schur functions, St. Petersburg Math. J., 9 (1998).

[12] S. Sahi, Interpolation, integrality, and a generalization of Macdonald's polynomials, Internat. Math. Res. Notices, 10 (1996), 457-471. | MR 99j:05189b | Zbl 0861.05063