Basic Hypergeometric Series - The q-binomial Theorem

The q-binomial Theorem

The q-binomial theorem (first published in 1811 by Heinrich August Rothe) states that

\;_{1}\phi_0 (a;q,z) =\frac{(az;q)_\infty}{(z;q)_\infty}= \prod_{n=0}^\infty \frac {1-aq^n z}{1-q^n z}

which follows by repeatedly applying the identity

\;_{1}\phi_0 (a;q,z) = \frac {1-az}{1-z} \;_{1}\phi_0 (a;q,qz).

The special case of a = 0 is closely related to the q-exponential.

Read more about this topic:  Basic Hypergeometric Series

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