Symmetric and nonsymmetric Koornwinder polynomials in the \(q \to 0\) limit
Koornwinder polynomials, orthogonal polynomials, symmetric functions are some of the well known key words that are included in the present paper, which analyses the problem in two parts or sections. Macdonald 2000/2001 introduced a family of multivariate q-orthogonal polynomials associated with a root system. It is known, see reference [\textit{I. G. Macdonald}, Sémin. Lothar. Comb. 45, B45a, 40 p. (2000; Zbl 1032.33010)] of the present paper, that for the type A root system, the Macdonald polynomials contain many well known families of symmetric functions, viz. Schur, Hall-Littlewood among other. The Askey-Wilson polynomials under certain restrictions are a fundamental class of orthogonal polynomials associated with certain root system, see reference [\textit{R. Askey} and \textit{J. Wilson}, Mem. Am. Math. Soc. 319, 55 p. (1985; Zbl 0572.33012)] of the paper. In suitable limits of parameters, Macdonald polynomials can be recovered from Koornwinder polynomials. Interesting to observe is that similar to the case of Macdonald polynomials, the existence of some other polynomials, mentioned above and also on page 332 of the paper, was proved by using q-difference operators, but it is found that these behave badly as \(q \to 0\). A natural question, if a relation between Macdonald and Koornwinder polynomials is known (or given), then does there exists an explicit construction for the Koornwinder polynomials at \(q=0\)? The first part of the paper under review answers this. The second part of the paper deals with the non-symmetric theory for the Koornwinder polynomials in the limit \(q \to 0\), in the Askey-Wilson parametrization. An explicit formula is given when these polynomials are indexed by partitions. It may be noted that the third section of the first part of the paper contains an application of the main formula. In Theorem 2.8, on page 343 of this paper, one finds a direct proof of Gustafson's formula in the limit \(q=0\). The paper is nicely composed and the analysis is simple and elegant.
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