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.
There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic geometric series ψ. The unilateral basic hypergeometric series is defined as
is the q-shifted factorial. The most important special case is when j = k+1, when it becomes
This series is called balanced if a1...ak+1 = b1...bkq. This series is called well poised if a1q = a2b1 = ... = ak+1bk, and very well poised if in addition a2 = −a3 = qa11/2.
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as
The most important special case is when j = k, when it becomes
The unilateral series can be obtained as a special case of the bilateral one by setting one of the b variables equal to q, at least when none of the a variables is a power of q., as all the terms with n<0 then vanish.
Some simple series expressions include
The q-binomial theorem
which follows by repeatedly applying the identity
The special case of a = 0 is closely related to the q-exponential.
Ramanujan gave the identity
valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for have been given by Bailey. Such identities can be understood to be generalizations of the Jacobi triple product theorem, which can be written using q-series as
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. There is a similar contour integral for r+1φr. This contour integral gives an analytic continuation of the basic hypergeometric function in z.
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