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

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

when |q|<1, and

\( \Gamma_q(x)=\frac{(q^{-1};q^{-1})_\infty}{(q^{-x};q^{-1})_\infty}(q-1)^{1-x}q^{\binom{x}{2}} \)

if |q|>1. Here (·;·)∞ is the infinite q-Pochhammer symbol. It satisfies the functional equation

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

For non-negative integers n,

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

where [·]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

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

A q-analogue of Stirling's formula for |q|<1 is given by

\( \Gamma_q(x) =[2]_{q^{\ }}^{\frac 12} \Gamma_{q^2}\left(\frac 12\right)(1-q)^{\frac 12-x}e^{\frac{\theta q^x}{1-q-q^x}}, \quad 0<\theta<1. \)

A q-analogue of the multiplication formula for |q|<1 is given by

\( \Gamma_{q^n}\left(\frac {x}n\right)\Gamma_{q^n}\left(\frac {x+1}n\right)\cdots\Gamma_{q^n}\left(\frac {x+n-1}n\right) =[n]_q^{\frac 12-x}\left([2]_q \Gamma^2_{q^2}\left(\frac12\right)\right)^{\frac{n-1}{2}}\Gamma_q(x). \)

Due to I. Mező, the q-analogue of the Raabe formula exists, at least if we use the q-gamma function when |q|>1. With this restriction

\( \int_0^1\log\Gamma_q(x)dx=\frac{\zeta(2)}{\log q}+\log\sqrt{\frac{q-1}{\sqrt[6]{q}}}+\log(q^{-1};q^{-1})_\infty \quad(q>1). \)

References

Jackson, F. H. (1905), "The Basic Gamma-Function and the Elliptic Functions", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society) 76 (508): 127–144, doi:10.1098/rspa.1905.0011, ISSN 0950-1207, JSTOR 92601
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
Mansour, M (2006), "An asymptotic expansion of the q-gamma function Γq(x)", Journal of Nonlinear Mathematical Physics 13 (4): 479–483, doi:10.2991/jnmp.2006.13.4.2[1]
Mező, István (2012), "A q-Raabe formula and an integral of the fourth Jacobi theta function", Journal of Number Theory 130 (2): 360–369

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