On the Normalizing Multiplier of the Generalized Jackson Kernel

We consider the question of evaluating the normalizing multiplier
$$ \gamma_{n,k} = \frac1 \pi \int_{-\pi}^\pi {\biggl(\frac{\sin\frac{n t}2}{\sin\frac t 2}\biggr)}^{2k}\,dt $$
for the generalized Jackson kernel $J_{n,k}(t)$. We obtain the explicit formula
$$ \gamma_{n,k} = 2 \sum_{p=0}^{[k-\frac k n]} (-1)^p \binom{2k}p \binom{k(n+1) - np - 1}{k(n-1) - np} $$
and the representation
$$ \gamma_{n,k} = \sqrt{\frac{24}{\pi}}\cdot\frac {(n-1)^{2k-1}}{\sqrt{2k-1}}\left[ 1 - \frac 1{8}\cdot\frac{1}{2k-1} + \omega(n,k)\right], $$
where
$$ |{\omega(n,k)}|<\frac{4}{(2k-1)\sqrt{\ln(2k-1)}}+ \sqrt{12\pi}\cdot\frac{k^\frac{3}{2}}{n-1}\left(1+ \frac{1}{n-1}\right)^{2k-2}. $$