Basic Hypergeometric Series - Definition

Definition

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

\;_{j}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{j} \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=0}^\infty \frac {(a_1, a_2, \ldots, a_{j};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n

where

and where

is the q-shifted factorial. The most important special case is when j = k+1, when it becomes

\;_{k+1}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{k}&a_{k+1} \\ b_1 & b_2 & \ldots & b_{k} \end{matrix} ; q,z \right] = \sum_{n=0}^\infty \frac {(a_1, a_2, \ldots, a_{k+1};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} z^n.

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

\;_j\psi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a_1, a_2, \ldots, a_j;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} \left((-1)^nq^{n\choose 2}\right)^{k-j}z^n.

The most important special case is when j = k, when it becomes

\;_k\psi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_k \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a_1, a_2, \ldots, a_k;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} z^n.

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.

Read more about this topic:  Basic Hypergeometric Series

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