Hypergeometric function information

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

a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series...

a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential...

functions can be expressed in terms of the gamma function. More functions yet, including the hypergeometric function and special cases thereof, can be represented...

random variable X {\displaystyle X} follows the hypergeometric distribution if its probability mass function (pmf) is given by p X ( k ) = Pr ( X = k ) =...

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

Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function): erf ⁡ x = 2 x π M ( 1 2 , 3 2 , − x 2 ) . {\displaystyle...

The exponential function is a mathematical function denoted by f ( x ) = exp ⁡ ( x ) {\displaystyle f(x)=\exp(x)} or e x {\displaystyle e^{x}} (where...

{1}{(1-t)^{\alpha +1}}}e^{-tx/(1-t)}.} Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as L n ( α ) ( x...

by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the...

function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function...

hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions...

Gustav Jacob Jacobi. The Jacobi polynomials are defined via the hypergeometric function as follows: P n ( α , β ) ( z ) = ( α + 1 ) n n ! 2 F 1 ( − n ...

{z^{s+k}}{s+k}}={\frac {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...

expressed in terms of the hypergeometric function, 2 F 1 {\displaystyle _{2}F_{1}} . With Γ {\displaystyle \Gamma } being the gamma function, the first solution...

elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series...

generalization resembles the hypergeometric function and the Meijer G function but it belongs to a different class of functions. When r1 = r2, both sides...

the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial...

is the gamma function and 2 F 1 ( a , b ; c ; z ) {\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)} is the Gaussian hypergeometric function. In the special...

{\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 (...

}e^{-x\sinh t-\alpha t}\,dt.} The Bessel functions can be expressed in terms of the generalized hypergeometric series as J α ( x ) = ( x 2 ) α Γ ( α +...

mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced...

the time, such as the elementary functions and some special functions, are a special case of the hypergeometric function. This work is the first one with...

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

mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These...

{1}{2}}}(1/x)} The Bessel polynomial may also be defined as a confluent hypergeometric function: 8  y n ( x ) = 2 F 0 ( − n , n + 1 ; ; − x / 2 ) = ( 2 x ) − n...

{i^{l}}{(m+nl+1)}}{\frac {x^{m+nl+1}}{l!}}} is a confluent hypergeometric function and also an incomplete gamma function ∫ x m e i x n d x = x m + 1 m + 1 1 F 1 ( m...

hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function....