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...

equation Generalized hypergeometric functions, which generalize the hypergeometric function to specific higher orders General hypergeometric functions, which...

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...

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

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...

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

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

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

}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 +...

Pure Appl. Math. 5: 337–361. Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge University Press. pp. 42–44. ISBN 978-0-521-06483-5...

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

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...

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

{z^{2}}{4}}),} where pFq is a generalized hypergeometric function. Anger function Lommel polynomial Struve function Weber function Watson's "Treatise on 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...

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

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

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...

{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...

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

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...

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

immediately gives rise to an expression in terms of the generalized hypergeometric function 2 F 2 {\displaystyle {}_{2}F_{2}} : Ei ⁡ ( x ) = x 2 F 2...

this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function: K v ( x , y ) = ∫ 1 ∞ e − x t − y t t v...

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 ) = (...