Koornwinder polynomials

introduction

• The Koornwinder polynomials [43] are a generalisation of the Macdonald polynomials to the root system BCn.
• They depend on six parameters, except for n = 1 when they correspond to the 5-parameter Askey–Wilson polynomials [3].

history

• Several years after the work of Askey and Wilson, Koornwinder extended the Askey–Wilson polynomials to a family of multivariable Laurent polynomials labelled by the non-reduced root system $BC_n$
• The various families of Macdonald (orthogonal) polynomials for classical root systems are all contained in the Koornwinder polynomials, and for a long time it was assumed they represented the highest possible level of generalisation.

encyclopedia

• https://en.wikipedia.org/wiki/Koornwinder_polynomials

articles

• Corteel, Sylvie, and Lauren Williams. ‘Macdonald-Koornwinder Moments and the Two-Species Exclusion Process’. arXiv:1505.00843 [cond-Mat, Physics:nlin], 4 May 2015. http://arxiv.org/abs/1505.00843.
• Stokman, Jasper V. “Lecture Notes on Koornwinder Polynomials.” In Laredo Lectures on Orthogonal Polynomials and Special Functions, 145–207. Adv. Theory Spec. Funct. Orthogonal Polynomials. Nova Sci. Publ., Hauppauge, NY, 2004. http://www.ams.org/mathscinet-getitem?mr=2085855.
• Stokman, Jasper V. “Macdonald-Koornwinder Polynomials.” arXiv:1111.6112 [math], November 25, 2011. http://arxiv.org/abs/1111.6112.
• [43] T. H. Koornwinder, Askey–Wilson polynomials for root systems of type BC in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, pp. 189–204, Contemp. Math. 138, Amer. Math. Soc., Providence, 1992.
• [3] R. Askey and J. A. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. Vol. 319, AMS, Providence, RI, 1985.