Basic hypergeometric series

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In mathematics, Heine's basic hypergeometric series, or hypergeometric q-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 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.


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



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 (Koekoek & Swarttouw (1996)).
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.

Simple series[edit]

Some simple series expressions include



The q-binomial theorem[edit]

The q-binomial theorem (first published in 1811 by Heinrich August Rothe)[1][2] 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[edit]

Cauchy binomial theorem is a special case of the q-binomial theorem[3].

Ramanujan's identity[edit]

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

Ken Ono gives a related formal power series

Watson's contour integral[edit]

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.

See Also[edit]


  1. ^ Bressoud, D. M. (1981), "Some identities for terminating q-series", Mathematical Proceedings of the Cambridge Philosophical Society, 89 (2): 211–223, MR 600238, doi:10.1017/S0305004100058114 .
  2. ^ Benaoum, H. B., "h-analogue of Newton's binomial formula", Journal of Physics A: Mathematical and General, 31 (46): L751–L754, arXiv:math-ph/9812011Freely accessible, doi:10.1088/0305-4470/31/46/001 .
  3. ^ Wolfram Mathworld: Cauchy Binomial Theorem

External links[edit]


  • Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions, Encyclopedia of Mathematics and its Applications, volume 71, Cambridge University Press.
  • Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.
  • Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin.