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The transformed Macdonald polynomials ~ (;,) in the formula above are related to the classical Macdonald polynomials via a sequence of transformations. First, the integral form of the Macdonald polynomials, denoted J λ ( x ; q , t ) {\displaystyle J_{\lambda }(x;q,t)} , is a re-scaling of P λ ( x ; q , t ) {\displaystyle P_{\lambda }(x;q,t ...
The following implementation in the Python programming language works for complex arguments and typically gives 13 correct decimal places. Note that omitting the smallest coefficients (in pursuit of speed, for example) gives totally inaccurate results; the coefficients must be recomputed from scratch for an expansion with fewer terms.
In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. [1] Note that such factors may well be functions of the parameters of the
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 .
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 ...
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...
Unlike Kummer's function which is an entire function of z, U(z) usually has a singularity at zero. For example, if b = 0 and a ≠ 0 then Γ(a+1)U(a, b, z) − 1 is asymptotic to az ln z as z goes to zero. But see #Special cases for some examples where it is an entire function (polynomial).
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.