Please allow me to resort once again to the expertise of the MathOverflow community :

During research I encoutered the following infinite series :

$$\sum_{n=-\infty}^{+\infty} \frac{u^{2n}}{1+\rho^{2n+1}},$$ where $0<\rho<1$,$-1<u<1$ and $\rho<|u|$.

Is this series related to some well-known special functions, for instance the elliptic functions or theta functions? I have reasons to believe that it should be.

Thank you,

Best regards, Malik

**Note**
Unless I made a mistake, the above sum can be rewritten as
$$\sum_{m=0}^{\infty} (-1)^m \frac{\rho^m}{1-u^2\rho^{2m}} + \sum_{m=0}^{\infty} (-1)^m \frac{u^{-2}\rho^{m+1}}{1-u^{-2}\rho^{2m+2}},$$
which suspicously is quite similar to the infinite series appearing in the formulas for the expansion of the Weierstrass elliptic functions in term of the elliptic nome (see e.g. Akhiezer's Elements of the theory of elliptic functions, p.204).

**EDIT** (In response to Alexandre Eremenko's comment)

The series is convergent. Indeed, the $n<0$ part can be written

$$\sum_{n=1}^{\infty} \frac{u^{-2n}}{1+\rho^{-2n+1}} = \sum_{n=1}^{\infty} \frac{\rho^{2n}u^{-2n}}{\rho^{2n}+\rho}$$ which is clearly convergent since $\rho<|u|$.