Hypergeometric function of a matrix argument information
In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals.
Hypergeometric functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are often expressed in terms of the hypergeometric function of a matrix argument.
and 26 Related for: Hypergeometric function of a matrix argument information
ordinary hypergeometricfunction 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as...
In mathematics, the E-function was introduced by Thomas Murray MacRobert (1937–1938) to extend the generalized hypergeometric series pFq(·) to the case...
exponential function is a mathematical function denoted by f ( x ) = exp ( x ) {\displaystyle f(x)=\exp(x)} or e x {\displaystyle e^{x}} (where the argument x...
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced...
D. St. P. Richards (n.d.). "Chapter 35 FunctionsofMatrixArgument". Digital Library of Mathematical Functions. Retrieved 23 July 2022. Andrews, George...
{z^{s}}{s}}M(s,s+1,-z),} where M is Kummer's confluent hypergeometricfunction. When the real part of z is positive, γ ( s , z ) = s − 1 z s e − z M ( 1 ...
(occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are a class of classical orthogonal polynomials...
In the physical sciences, the Airy function (or Airy functionof the first kind) Ai(x) is a special function named after the British astronomer George...
{1}{2}};x^{2})} where 1 F 1 ( a ; b ; z ) {\displaystyle {}_{1}F_{1}(a;b;z)} are Confluent hypergeometricfunctionsof the first kind. The conventional...
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients ofa formal power series. Unlike an ordinary...
polynomials reduce to the Chebyshev polynomials of the second kind. They are given as Gaussian hypergeometric series in certain cases where the series is...
expressed in terms of hypergeometricfunctions. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable t = cos...
exponent ofmatrix multiplication is 2. Algorithms for computing transforms offunctions (particularly integral transforms) are widely used in all areas of mathematics...
characteristic functionof the beta distribution to a Bessel function, since in the special case α + β = 2α the confluent hypergeometricfunction (of the first...
probability function P {\displaystyle P} can take as argument subsets of the sample space itself, as in the coin toss example, where the function P {\displaystyle...
\left(-a^{2}r^{2}\right)J_{0}(kr)=M\left(n+1,1,-{k^{2} \over 4a^{2}}\right).} Here, M is a confluent hypergeometricfunction. For an application of this...
distribution Hypergeometric distribution Hyperparameter Hyperprior Hypoexponential distribution Idealised population Idempotent matrix Identifiability...
fusing matrix, the integral is a hyperbolic Barnes integral. Up to normalization, the fusing matrix coincides with Ruijsenaars' hypergeometricfunction, with...
version of the hypergeometric differential equation Curiously, they have been omitted from the standard textbooks on special functions in mathematical...
{\displaystyle \Gamma } is the gamma function and 2 F 1 {\displaystyle _{2}F_{1}} is the hypergeometricfunction 2 F 1 ( α , β ; γ ; z ) = Γ ( γ ) Γ (...
estimate Bessel functions and pointed out that it occurred in the unpublished note by Riemann (1863) about hypergeometricfunctions. The contour of steepest...