Basic hypergeometric function

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 x_n is called hypergeometric if the ratio of successive terms x_n+1/x_n 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(q^α,q^β;q^γ;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

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 ... a_{k + 1} = b₁ ...b_kq. This series is called well poised if a₁q = a₂b₁ = ... = a_{k + 1}b_k, and very well poised if in addition a₂ = −a₃ = qa₁^1/2.

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.

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