q-gamma function

In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary Gamma function closely related to the double gamma function. It was introduced by Jackson (1905). It is given by

\Gamma_q(x) = (1-q)^{1-x}\prod_{n=0}^\infty \frac{1-q^{n%2B1}}{1-q^{n%2Bx}}=(1-q)^{1-x}\,\frac{(q;q)_\infty}{(q^x;q)_\infty}

where (\cdot;\cdot)_\infty is the infinite q-Pochhammer symbol. It satisfies the functional equation

\Gamma_q(x%2B1) = \frac{1-q^{x}}{1-q}\Gamma_q(x)=[x]_q\Gamma_q(x)

For non-negative integers n,

\Gamma_q(n)=[n-1]_q!

where [\cdot]_q! is the q-factorial function. Alternatively, this can be taken as an extension of the q-factorial function to the real number system.

The relation to the ordinary gamma function is made explicit in the limit

\lim_{q \to 1} \Gamma_q(x) = \Gamma(x).

References