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Unfortunately, the referenced article 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 gives wrong formulae
It is discussed in detail in Koepf, Wolfram; Rajković, Predrag M.; Marinković, Sladjana D. (2016), "On a connection between formulas about q–gamma functions", Journal of Nonlinear Mathematical Physics, 23 (3), doi:10.1080/14029251.2016.1199496, and can be easily checked numerically.
![{\displaystyle \Gamma _{q}(x)=[2]_{q^{\ }}^{\frac {1}{2}}\Gamma _{q^{2}}\left({\frac {1}{2}}\right)(1-q)^{{\frac {1}{2}}-x}e^{\frac {\theta q^{x}}{1-q-q^{x}}},\quad 0<\theta <1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9e8d25291b18f7be7dba6c3eb740809d5ca997)
![{\displaystyle \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 {1}{2}}-x}\left([2]_{q}\Gamma _{q^{2}}^{2}\left({\frac {1}{2}}\right)\right)^{\frac {n-1}{2}}\Gamma _{q}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1dca8fdc67944b3be5b715e7d36401358f5ace)