相关论文: A nonsymmetric version of Okounkov's BC-type inter…
In this elementary paper we prove that the extra vanishing property characterizes the BC interpolation Macdonald polynomials inside a very general class of multivariate interpolation polynomials. It follows that they are the only…
In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial…
We investigate the connections between various noncommutative analogues of Hall-Littlewood and Macdonald polynomials, and define some new families of noncommutative symmetric functions depending on two sequences of parameters.
In symmetric Macdonald polynomial theory the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function and the Macdonald polynomial. In this paper we give the nonsymmetric analogues for the…
The symmetric Macdonald polynomials are able to be constructed out of the non-symmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their non-symmetric…
Bisymmetric Macdonald polynomials can be obtained through a process of antisymmetrization and $t$-symmetrization of non-symmetric Macdonald polynomials. Using the double affine Hecke algebra, we show that the evaluation of the bisymmetric…
We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition. These are polynomials which are…
We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bessel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Pad\'e table. We give some properties of…
In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ and integral Macdonald polynomials $J_{\lambda}(X;q,t)$, in terms of several new statistics and the major index for a partition…
We present a new formula of Cauchy type for the nonsymmetric Macdonald polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the explicit formula for a reproducing kernel on the polynomial ring of n variables.
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters $(q,t)$ and polynomial in a further two parameters $(u,v)$. We evaluate these polynomials explicitly as a matrix…
We construct a family of BC_n-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials, and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based…
Let Sym denote the algebra of symmetric functions and $P_\mu(\,\cdot\,;q,t)$ and $Q_\mu(\,\cdot\,;q,t)$ be the Macdonald symmetric functions (recall that they differ by scalar factors only). The $(q,t)$-Cauchy identity $$ \sum_\mu…
Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the…
The affine Hecke algebra of type $A$ has two parameters $\left( q,t\right) $ and acts on polynomials in $N$ variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys-Murphy…
Symmetric Jack polynomials arise naturally in several contexts, including statistics, physics, combinatorics, and representation theory. They are pairwise orthogonal with repsect to two different inner products, the first defined by…
The Macdonald polynomials with prescribed symmetry are obtained from the nonsymmetric Macdonald polynomials via the operations of $t$-symmetrisation, $t$-antisymmetrisation and normalisation. Motivated by corresponding results in Jack…
An algorithm for generating interpolants for formulas which are conjunctions of quadratic polynomial inequalities (both strict and nonstrict) is proposed. The algorithm is based on a key observation that quadratic polynomial inequalities…
Starting from nonlocal symmetries related to B\"acklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact…
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…