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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. If we put t = q α and let q tend to 1 the Macdonald polynomials become Jack polynomials when the root system is of type A , and Heckman–Opdam polynomials for more general root systems.
Macdonald function after Hector Munro Macdonald; Spherical Bessel functions: j n, y n. Plot of the spherical Bessel function of the first kind j n (z) with n = 0.5 in ...
This is a list of special function eponyms in mathematics, ... Ian G. Macdonald: Macdonald polynomial, Macdonald–Kostka polynomial, Macdonald spherical function.
Macdonald at Oberwolfach in 1977. Ian Grant Macdonald FRS (11 October 1928 – 8 August 2023) was a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinatorics.
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.
In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald . They include as special cases the Jacobi triple product identity , Watson's quintuple product identity , several identities found by Dyson (1972) , and a 10-fold product identity found by Winquist (1969) .
A ring of symmetric functions can be defined over any commutative ring R, and will be denoted Λ R; the basic case is for R = Z. The ring Λ R is in fact a graded R-algebra. There are two main constructions for it; the first one given below can be found in (Stanley, 1999), and the second is essentially the one given in (Macdonald, 1979).
In mathematics, the Kontorovich–Lebedev transform is an integral transform which uses a Macdonald function (modified Bessel function of the second kind) with imaginary index as its kernel. Unlike other Bessel function transforms, such as the Hankel transform, this transform involves integrating over the index of the function rather than its ...