Q-gamma function

In q-analog theory, the ${\displaystyle q}$-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

when ${\displaystyle |q|<1}$, and

if ${\displaystyle |q|>1}$. Here ${\displaystyle (\cdot ;\cdot )_{\infty }}$ is the infinite q-Pochhammer symbol. The ${\displaystyle q}$-gamma function satisfies the functional equation

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