1
$\begingroup$

I have encountered the following sum involving Legendre polynomials, which I hope to reduce to something involving a $\delta$-function: $$ \frac{d^2}{dx^2} \sum_{\ell=0}^{\infty} \frac{2 \ell + 1}{2 \ell (\ell+1)} P_{\ell}(x) P_{\ell}(y) $$

I have played around with the recurrence relation for derivatives of Legendre polynomials without much success ($(1-x^2) P_{\ell}'(x) = -\ell x P_{\ell}(x) + \ell P_{\ell-1}(x)$). I have done some numerical investigation to try and gain intuition, and it does appear that the result vanishes for $x \neq y$, as expected for something proportional to $\delta(x-y)$. However, I've run into tricky divergences and other problems for the $x=y$ case.

If anyone has come across a sum or identity similar to this, I would be most obliged.

$\endgroup$

1 Answer 1

1
$\begingroup$

Firstly, I assume that $l$ goes from $1$ to infinity, otherwise you are dividing by $0.$ Secondly, without the derivative, mathematica says that $\sum_{l=1}^\infty \frac{2l+1}{l(l+1)} P_l(x) P_l(y) = $

if $x < y,$ $ -\log (-(x-1) (y+1))+2 \log (2)-1$,

if $x > y$ then $ -\log (-(x+1) (y-1))+2 \log (2)-1$

and $0$ otherwise.

You can differentiate this twice and see what you get.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.