I am interested in computing the following sum:
\begin{equation} \sum\limits_{l=1}^k l^{\beta_1} \cos\left(\omega \log_q(\frac{l}{t_c})\right) \end{equation}
Here $0 < \omega$, $0 < k < t_c$, $q \in (0,2)$ and $\ln_q(x) := \frac{x^{1-q} - 1}{1-q}$ is the generalization of the natural logarithm.
Now, we know that if $q=1$ then the q-logarithm reduces to the logarithm, and using Euler's formula we can express the result through harmonic numbers as:
\begin{equation} \mathrm{Re}\left[ \frac{H_k^{\beta_1 + \imath \omega}}{t_c^{\imath \omega}} \right] \end{equation}
Since Harmonic numbers are easily computed in Mathematica or in Matlab the problem is solved. However, how shall I handle the generic case $q\ne 1$?
One way I wanted to tackle this problem is to use Euler's formula and reduce that sum to a following :
\begin{equation} \sum\limits_{l=0}^k l^{\beta_1} x^{l^{1-q}} \end{equation} Yet, since I heard that these kind of sums cannot be expressed in closed form, ie by means of elementary functions I am in a dead end. Any thoughts will be appreciated.