A series of the form
Such a series is meaningful if is not equal to zero or to a negative integer; it converges for . If, in addition, , it also converges for . In such a case the Gauss formula
where is the gamma-function, holds. An analytic function defined with the aid of a hypergeometric series is said to be a hypergeometric function.
A generalized hypergeometric series is a series of the form
where . If this notation is used, the series
is written as .
Generalized hypergeometric series can be characterized as power series such that is a rational function of . An analogous characterization for series in two variables was given by J. Horn. This yields a class of power series in two variables which includes the various Appell's hypergeometric series, cf. [a1].
Basic hypergeometric series can be characterized as power series such that is a rational function of , where is a fixed complex number not equal to 0 or 1. Such series have the form
where . See [a2].
Hypergeometric functions of matrix argument have also been studied, cf. [a3].
|[a1]||P. Appell, M.J. Kampé de Fériet, "Fonctions hypergéométriques et hypersphériques: Polynômes d'Hermite" , Gauthier-Villars (1926)|
|[a2]||G. Gasper, M. Rahman, "Basic hypergeometric series" , Cambridge Univ. Press (1989)|
|[a3]||K. Gross, D. Richards, "Special functions of matrix argument I" Trans. Amer. Math. Soc. , 301 (1987) pp. 781–811|
Hypergeometric series. E.A. Chistova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hypergeometric_series&oldid=11231