# Basic Hypergeometric Series

In mathematics, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base.

The basic hypergeometric series 2φ1(qα,qβ;qγ;q,x) was first considered by Eduard Heine (1846). It becomes the hypergeometric series F(α,β;γ;x) in the limit when the base q is 1.

Read more about Basic Hypergeometric Series:  Definition, Simple Series, The q-binomial Theorem, Ramanujan's Identity, Watson's Contour Integral

### Other articles related to "basic hypergeometric series, hypergeometric series, basic hypergeometric":

Basic Hypergeometric Series - Watson's Contour Integral
... As an analogue of the Barnes integral for the hypergeometric series, Watson showed that where the poles of lie to the left of the contour and the remaining poles lie to the right ... This contour integral gives an analytic continuation of the basic hypergeometric function in z ...

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