Summation of Legendre polynomial series

Question

How do I find the sum $$\sum_{n=0}^{\infty}(-1)^n P_n(x)$$ where $P_n$ are the $n$th order Legendre polynomials? I tried using the generating function but I was not able to arrive at an answer. Any hints appreciated.

Answer 1

I don't see any problem in using the generating function. $$\frac{1}{\sqrt{r^2-2rx+1}} = \sum_{n=0}^{\infty} P_n(x)r^n$$ holds for all $|r|<1$ (as the radius of convergence in the complex plane is the distance to the first singular point, which here lie both on the unit circle) and $|x| \le 1$ and it holds also for $r=-1$, your example. Thus $$\sum_{n=0}^{\infty} P_n(x)(-1)^n = \frac{1}{\sqrt{2+2x}}$$ As already remarked by ${conditionalMethod}$. Note that this leads to an ambigious result for $x=-1$.