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If q = 0 the Macdonald polynomials become the (rescaled) zonal spherical functions for a semisimple p-adic group, or the Hall–Littlewood polynomials when the root system has type A. If t=1 the Macdonald polynomials become the sums over W orbits, which are the monomial symmetric functions when the root system has type A.
Ian G. Macdonald: Macdonald polynomial, Macdonald–Kostka polynomial, Macdonald spherical function. Mahler polynomial; Maitland function; Émile Léonard Mathieu: Mathieu function
The two-variable Kostka polynomials K λμ (q, t) are known by several names including Kostka–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials. Here the indices λ and μ are integer partitions and K λμ ( q , t ) is polynomial in the variables q and t .
Any symmetric polynomial in n indeterminates can be used to construct a compatible family of symmetric polynomials, using the homomorphisms ρ i for i < n to decrease the number of indeterminates, and φ i for i ≥ n to increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained by symmetry from ...
The affine root system of type G 2.. In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space.They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-adic algebraic groups, and correspond to families of Macdonald polynomials.
The Macdonald polynomials are a two-parameter family of orthogonal polynomials indexed by a positive weight λ of a root system, introduced by Ian G. Macdonald (1987). They generalize several other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials.
Often, our pets don’t get to make their own choices, as we decide when they need to go to the vet and when we feed them, for example. So, giving them choices during training sessions can go a ...
In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous , symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials .