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

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...

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 ) =...

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...

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...

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 those as...

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...

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

{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...

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

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

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

{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...

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

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...

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 ...

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

) {\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)} is the confluent hypergeometric function. Other pairs of independent solutions may be formed from linear...

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...

}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 +...

{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...

involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and much simpler sum...

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

Appell polynomials can be obtained in terms of the generalized hypergeometric function. Let Δ ( k , − n ) {\displaystyle \Delta (k,-n)} denote the array...

the general hypergeometric function F ( α , β , γ , x ) {\displaystyle F(\alpha ,\beta ,\gamma ,x)} , and shows that many of the functions known at the...

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....

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...

are increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and...