2
$\begingroup$

Legendre polynomials can be given by several expressions, but perhaps the most compact way to represent them is by Rodrigues' formula as

$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n.$$

I was wondering if there exists a multidimensional variant of the Legendre polynomials, e.g. an analogous $P_n(x,y,z)$ for instance. Does anyone know anything about this? I can't seem to find anything about it.

$\endgroup$
1
  • $\begingroup$ I mean usually when you expand $\dfrac{1}{|\mathbf x-\mathbf x'|}$ you find a coefficient of the form $$P_\ell(\mathbf{\hat x}\cdot\mathbf{\hat x}')=P_\ell\left(\frac{xx'+yy'+zz'}{|\mathbf x|\,|\mathbf x'|}\right)$$ which depends on multiple variables kinda $\endgroup$
    – Conreu
    Commented 22 hours ago

0

Reset to default

You must log in to answer this question.