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

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

only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include...

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...

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

exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. The exponential of a variable...

{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 ...

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

univariate real-valued distributions, moment-generating functions can also be defined for vector- or matrix-valued random variables, and can even be extended...

mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are...

becomes a function solely of the model parameters. In maximum likelihood estimation, the argument that maximizes the likelihood function serves as a point...

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

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

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

{\mathrm {d} t}{\sqrt {1-t^{4}}}}.} It can also be represented by the hypergeometric function: arcsl ⁡ x = x 2 F 1 ( 1 2 , 1 4 ; 5 4 ; x 4 ) {\displaystyle \operatorname...

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

{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...

statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical...

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

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...

tests. As a function of ⁠ p {\displaystyle p} ⁠, the sum of this series is Riemann's zeta function. Hypergeometric series: p F q [ a 1 , a 2 , … , a p b 1...

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...

Random effects model Random element Random field Random function Random graph Random matrix Random measure Random multinomial logit Random naive Bayes...

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...

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

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

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

correlation, total positivity, and hypergeometric functions of matrix argument. He currently serves as a distinguished professor of statistics at the Pennsylvania...

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

\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...