Trigonometric Padé approximants of special functions

For the functions $H_\gamma=\sum_{k=1}^\infty\sin kx/(\gamma)_k$, where $(\gamma)_k=\gamma(\gamma+1)\cdots(\gamma+k-1)$ and their trigonometric Padé approximations $\pi^t_{n,m}(x;H_\gamma)$ the asymptotics of decreasing difference $H_\gamma(x)-\pi^t_{n,m}(x;H_\gamma)$ in the case is found, where $0\leqslant m\leqslant m(n)$, $m(n)=o(n)$, as $n\to\infty$. Particulary, we determine that, under the same assumption, the trigonometric Padé approximations $\pi^t_{n,m}(x;H_\gamma)$ converge to $H_\gamma$ uniformly on the $\mathbb{R}$ with the asymptotically best rate.