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In 1988, I.G. Macdonald [2] gave the second proof of a combinatorial interpretation of the Macdonald polynomials (equations (4.11) and (5.13)). Macdonald’s formula is different to that in Haglund, Haiman, and Loehr's work, with many fewer terms (this formula is proved also in Macdonald's seminal work, [ 3 ] Ch. VI (7.13)).
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 ...
Qalculate! supports common mathematical functions and operations, multiple bases, autocompletion, complex numbers, infinite numbers, arrays and matrices, variables, mathematical and physical constants, user-defined functions, symbolic derivation and integration, solving of equations involving unknowns, uncertainty propagation using interval arithmetic, plotting using Gnuplot, unit and currency ...
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.
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).
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
The Macdonald function (Modified Bessel function of the II kind) (Abramowitz and Stegun, 1972, p.376) is defined by: ...
The free-space circular cylindrical Green's function (see below) is given in terms of the reciprocal distance between two points. The expression is derived in Jackson's Classical Electrodynamics. [1] Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in