Hypergeometric Series

(redirected from Gauss hypergeometric function)

hypergeometric series

[‚hī·pər‚jē·ə′me·trik ′sir·ēz]
(mathematics)
A particular infinite series which in certain cases is a solution to the hypergeometric equation, and having the form:

Hypergeometric Series

 

a series of the form

F(α, β; γ; z) = 1

Hypergeometric Series

The hypergeometric series was studied for the first time by L. Euler in 1778. Expansions of many functions into infinite series are special cases of hypergeometric series. For example,

Hypergeometric Series

The hypergeometric series has meaning if γ is not equal to zero or a negative integer. It converges at |z| < 1. If, in addition, γ - α - β > 0, then the hypergeometric series also converges for z = 1. In this case, Gauss’ formula is valid:

Hypergeometric Series

where Γ(z) is the gamma function. An analytic function, defined for |z| < 1 by a hypergeometric series is called a hypergeometric function and plays an important role in the theory of differential equations.

References in periodicals archive ?
In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as
It is known as Gauss hypergeometric function in terms of Pochhammer symbol [(a).
1] Gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in the evaluation of [sup.
The Euler integral representation of the Gauss hypergeometric function, or [sub.
The results such as the cumulative distribution function, the mean and the variance of the beta type 3 random variable involve Gauss hypergeometric function which is easily computable using the software Mathematica.
Finally, using the integral representation of Gauss hypergeometric function (Abramowitz and Stegun[?
the Gauss hypergeometric function given in the above expression can be re-written as