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

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

Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023. (Srivastava...

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

hypergeometric distributions Negative hypergeometric distribution Multinomial distribution Sampling (statistics) Generalized hypergeometric function Coupon...

+1)z^{-\alpha /2}I_{\alpha }\left(2{\sqrt {z}}\right),} (see generalized hypergeometric function), this can also be written as ∑ n = 0 ∞ n ! Γ ( 1 + α + n...

hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by...

function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric...

product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The...

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 those...

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...

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

four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series...

current value of ⁠ f ( x ) {\displaystyle f(x)} ⁠. The exponential function can be generalized to accept complex numbers as arguments. This reveals relations...

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

}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 +...

of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function lists...

{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...

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

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...

introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by C n ( x ; μ ) = 2 F 0 ( − n , − x ; − ; − 1 / μ ) = (...

in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by S n ( x 2 ; a , b ,...

(corrigendum 1956 in Ganita 7, p. 65) Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X...

{\sqrt {1+z}}} , the dilogarithm function Li2(z), the generalized hypergeometric functions pFq(...; ...; z) and the functions defined by the power series ∑...

the class of hypergeometric functions. Examples of special functions that are holonomic but not hypergeometric include the Heun functions. Examples of...

} Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z): E 1 ( z ) = e − z U ( 1 , 1 ...

two terms is a rational function of n. The definition of the generalized hypergeometric series is similar, except that the terms with negative n must...

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...

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