Closed form of sum of infinite series of Legendre polynomials

Question

I was working on a research and we end up to discover that the Green's function on some domain is of the form :\begin{equation} G = \frac{-1}{4\pi}\sum_{l=0}^{\infty}\frac{2l+1}{l(l+1)+\frac{1}{\alpha}}P_l(x) \end{equation} where $\alpha$ $\in$ $\mathbb{R}^+$. I am seeking the closed form of this expression, is there any hints ? Thank you

Answer 1

We have $$ \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{l\geq 0} t^l P_l(x) \tag{1}$$ and if for any $\alpha>0$ we set $\beta_{\pm} = \frac{-\alpha\pm\sqrt{\alpha(\alpha-4)}}{2\alpha}$ we have $$ \frac{2l+1}{l(l+1)+\frac{1}{\alpha}}=\frac{1}{l-\beta_+}+\frac{1}{l-\beta_-} \tag{2}$$ so: $$ \sum_{l\geq 0}\frac{2l+1}{l(l+1)+\frac{1}{\alpha}}P_l(x) = \int_{0}^{1}\frac{dt}{t\sqrt{1-2xt+t^2}}\left(\frac{1}{t^{\beta_+}}+\frac{1}{t^{\beta_-}}\right)\,dt \tag{3}$$ and the RHS of $(3)$ can be estimated by studying the behaviour of the integrand function at its singular points $t=0$ and $t=x\pm\sqrt{x^2-1}$. If $\alpha=4$ we also have an explicit closed form depending on a complete elliptic integral of the first kind.