I am investigating a series of the form $$\sum_{n=0}^\infty \frac{1}{1 + e^{nx}}P_n(x)$$ where $P_n$ is the Legendre Polynomial of degree $n$.
Because I am dealing with many of these series, it would be very useful to have any closed form for the series. However, I have been unsuccessful in determining any possible closed forms. Is anyone able to recommend me any resources / closed forms that already exist from which this series could possibly be better understood?
I am under the impression the series will most likely involve elliptic functions of some sort, given that this resembles an Eisenstein series with the Legendre polynomials multiplied within, however, I am unable to find any resources that can help.
Even having a better understanding of the series $$\sum_{n=0}^\infty \text{sech}(nx)P_n(x)$$ in closed form would be very useful as well (in particular, for instance, this sum has a closed form without the Legendre polynomials, as shown in https://doi.org/10.1137/0510019, however I cannot find anything for series involving Legendre polynomials).
Any help would be appreciated!