The q-gamma function $\Gamma_q$ is defined as follows:
$\Gamma_q(x) = (1-q)^{1-x} \prod_{n=0}^\infty \frac{1-q^{n+1}}{1-q^{n+x}}$, when $|q|<1$. My question:
Could anyone help me calculate the following limit? $\displaystyle{\lim_{n\to\infty}}((x-m)\Gamma_q(n) + \sum_{k=1}^n \Gamma_q(k) - \Gamma_q (k+x-m)) $, for $m\in\mathbb{N}$ and $x\geq 0$.