Basic Hypergeometric Series

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...a_k+1 = b₁...b_kq. This series is called well poised if a₁q = a₂b₁ = ... = a_k+1b_k, 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

Famous quotes containing the word definition:

“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)

“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)

“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

Basic Hypergeometric Series - Definition

Famous quotes containing the word definition: