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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.

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Hypergeometric function of a matrix argument

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mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an...

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Hypergeometric function

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ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as...

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Generalized hypergeometric function

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if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation...

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MacRobert E function

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In mathematics, the E-function was introduced by Thomas Murray MacRobert (1937–1938) to extend the generalized hypergeometric series pFq(·) to the case...

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Exponential function

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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...

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Whittaker function

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In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced...

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Special functions

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D. St. P. Richards (n.d.). "Chapter 35 Functions of Matrix Argument". Digital Library of Mathematical Functions. Retrieved 23 July 2022. Andrews, George...

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Incomplete gamma function

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{z^{s}}{s}}M(s,s+1,-z),} where M is Kummer's confluent hypergeometric function. When the real part of z is positive, γ ( s , z ) = s − 1 z s e − z M ( 1 ...

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Jacobi polynomials

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(occasionally called hypergeometric polynomials) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are a class of classical orthogonal polynomials...

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Airy function

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In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George...

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Hermite polynomials

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{1}{2}};x^{2})} where 1 F 1 ( a ; b ; z ) {\displaystyle {}_{1}F_{1}(a;b;z)} are Confluent hypergeometric functions of the first kind. The conventional...

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Generating function

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In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Unlike an ordinary...

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Hurwitz zeta function

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a ) = Φ ( 1 , s , a ) . {\displaystyle \zeta (s,a)=\Phi (1,s,a).\,} Hypergeometric function ζ ( s , a ) = a − s ⋅ s + 1 F s ( 1 , a 1 , a 2 , … a s...

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Gegenbauer polynomials

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polynomials reduce to the Chebyshev polynomials of the second kind. They are given as Gaussian hypergeometric series in certain cases where the series is...

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List of dynamical systems and differential equations topics

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problem, many-body problem Ballistics Airy function Bessel function Legendre polynomials Hypergeometric function Angular velocity Angular momentum Angular...

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Mathieu function

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expressed in terms of hypergeometric functions. This can be seen by transformation of Mathieu's equation to algebraic form, using the change of variable t = cos...

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Computational complexity of mathematical operations

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exponent of matrix multiplication is 2. Algorithms for computing transforms of functions (particularly integral transforms) are widely used in all areas of mathematics...

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Beta distribution

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characteristic function of the beta distribution to a Bessel function, since in the special case α + β = 2α the confluent hypergeometric function (of the first...

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Probability distribution

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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...

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Common integrals in quantum field theory

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\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 hypergeometric function. For an application of this...

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List of statistics articles

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distribution Hypergeometric distribution Hyperparameter Hyperprior Hypoexponential distribution Idealised population Idempotent matrix Identifiability...

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Bingham distribution

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a confluent hypergeometric function of matrix argument. The matrices M and Z are the result of diagonalizing the positive-definite covariance matrix of...

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Virasoro conformal block

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fusing matrix, the integral is a hyperbolic Barnes integral. Up to normalization, the fusing matrix coincides with Ruijsenaars' hypergeometric function, with...

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Romanovski polynomials

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version of the hypergeometric differential equation Curiously, they have been omitted from the standard textbooks on special functions in mathematical...

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Associated Legendre polynomials

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{\displaystyle \Gamma } is the gamma function and 2 F 1 {\displaystyle _{2}F_{1}} is the hypergeometric function 2 F 1 ( α , β ; γ ; z ) = Γ ( γ ) Γ (...

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Method of steepest descent

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estimate Bessel functions and pointed out that it occurred in the unpublished note by Riemann (1863) about hypergeometric functions. The contour of steepest...

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