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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 ...
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
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).
Here is a sample program that computes the factorial of an integer number from 2 to 69 (ignoring the calculator's built-in factorial/gamma function). There are two versions of the example: one for algebraic mode and one for RPN mode. The RPN version is significantly shorter. Algebraic version:
A difference engine is an automatic mechanical calculator designed to tabulate polynomial functions. It was designed in the 1820s, and was created by Charles Babbage. The name difference engine is derived from the method of finite differences, a way to interpolate or tabulate functions by using a small set of polynomial co-efficients.
CORDIC (coordinate rotation digital computer), Volder's algorithm, Digit-by-digit method, Circular CORDIC (Jack E. Volder), [1] [2] Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), [3] [4] and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), [5] [6] is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots ...
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) .