Basic hypergeometric series

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.

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Related concepts (2)

Q-Pochhammer symbol

In mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with It is a q-analog of the Pochhammer symbol , in the sense that The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series.

Q-analog

In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century. q-analogs are most frequently studied in the mathematical fields of combinatorics and special functions.