# Hypergeometric series

*Gauss series*

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 .

#### Comments

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].

#### References

[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 |

**How to Cite This Entry:**

Hypergeometric series.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hypergeometric_series&oldid=11231