Some proprieties for the q-gamma functions via Wielandt's theorem (Q2352023)

The author presents the \(q\)-analogue of the Weierstrass product representation for the Jackson \(q\)-gamma function \[ \Gamma_q(z)=\frac{(q;q)_\infty}{(q^x;q)_\infty}(1-q)^{1-x}. \] This product -- which is also known as Gauss product formula -- reads as \[ \frac{1}{\Gamma_q(z)}=\frac{1}{(1-q)^{[z]_q-z}}[z]_qe^{\gamma_q[z]_q}\prod_{k=1}^\infty\frac{[z+k]_q}{[k]_q}e^{-\frac{q^k}{[k]_q}[z]_q}. \] The main tool is the \(q\)-analogue of Wielandt's theorem.