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Concept
Basic hypergeometric series
Summary
In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue 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 was first considered by . It becomes the hypergeometric series in the limit when base .
There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ.
The unilateral basic hypergeometric series is defined as
where
and
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 unilateral basic hypergeometric series is a q-analog of the hypergeometric series since
holds ().
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
and
and
The q-binomial theorem (first published in 1811 by Heinrich August Rothe) states that
which follows by repeatedly applying the identity
The special case of a = 0 is closely related to the q-exponential.
Cauchy binomial theorem is a special case of the q-binomial theorem.
Srinivasa Ramanujan gave the identity
valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for have been given by Bailey.
Official source
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