Hypergeometric Series


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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 ?
They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite
Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
Daoust, A note on the convergence of Kampe de Feriet doubles hypergeometric series, Math.
Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge.
Again Noor [4] by using hypergeometric series [sub.
In 1880 Paul Emile Appell (1855-1930) [8] introduced some 2-variable hypergeometric series now called the Appell functions.
However, if either or both of the numerator parameters a and b is zero or a negative integer, the hypergeometric series terminates.
BAILEY, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, Vol.
The theory of hypergeometric series and the theory of q-hypergeometric series will be united.
bq; z) is a function which can be defined in the form of a hypergeometric series, i.
Ponnusamy, Starlikeness properties for convolutions involving hypergeometric series, Ann.
q-Calculus is a generalization of many subjects, like hypergeometric series, complex analysis, and particle physics.