I'm working with the following series:
$$S_k = \sum_{j=1}^{p} \frac{\sqrt{1-r^{2|k-j|}}}{j} $$ here $|r|<1$, $k \leq p$.
Naturally, for small $r$ the series approaches harmonic sums $H_k \approx \gamma + \log(k) + 1/2k - 1/12k^2$, however, I'm trying to find a more manageable expression to work with..
By numerical experiments (guesswork mostly) I find that the following works reasonably well for $|r|<0.9$:
$$ S_k \approx H_p + \log\left(\frac{\beta_{11}}{1 + \exp\{\beta_{12} + \beta_{13} \cdot r\}}\right)\frac{1}{k} + e^{\beta_{21} + \beta_{22} r} \frac{1}{k^2} $$ for some numerical values $\beta_{11},\beta_{12},\beta_{13},\beta_{21},\beta_{22}$. Further, accuracy for $r$ closer to 1 can be improved by additional terms $k^{-3}$ and, probably, deeper expansion into $\text{exp}(\beta_{21} + \beta_{22} r + \beta_{23} r^2 + \ldots )$ near the $k^{-2}$ term.
All this suggests a possible expansion through $k^{-1}, k^{-2}, \ldots$, similar to approximation of $H_k$, just the summation weights will probably depend from $r$.
Does there exist any similar approximation that I could use analytically?
Just a few preliminary ideas (for the time being, I hope)
Using the expansion $$\sqrt{1-r^{2|k-j|}}=\sum_{n=0}^\infty (-1)^n \binom{\frac{1}{2}}{n}r^{2n|k-j|}$$ Let $$a_n=(-1)^n \binom{\frac{1}{2}}{n}r^{2n|k-j|}$$ $$\sum_{j=1}^k \frac{a_n}j=(-1)^{n+1} \binom{\frac{1}{2}}{n} \left(r^{-2 n}\Phi \left(r^{-2 n},1,k+1\right)+r^{2 k n} \log \left(1-r^{-2 n}\right)\right)$$ where appears the Lerch transcendent function.
Difficult to give now more than for "small" values of $r$ $$r^{-2 n}\Phi \left(r^{-2 n},1,k+1\right)\sim -\frac 1 {k+1}$$
Now, the problem is to work
$$r^{2k n} \log \left(1-r^{-2 n}\right)$$ whish seems to be more problematic
To follow, I hope and wish