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+1}}{1-q^{n+x}}=(1-q)^{1-x}\,\frac{(q;q)_\infty}{(q^x;q)_\infty}

where (·;·)∞ is the infinite q-Pochhammer symbol. It satisfies the functional equation

\Gamma_q(x+1) = \frac{1-q^{x}}{1-q}\Gamma_q(x)=_q\Gamma_q(x)

For non-negative integers n,

where 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

A q-analogue of Stirling's formula is given by

A q-analogue of the multiplication formula is given by

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