How to expand of the plane wave in Legendre polynomials

Question

There is the expression for the plane wave: $$ e^{i(\mathbf k \cdot \mathbf r )} = e^{ikrcos(\theta )} = e^{\frac{kr}{2}\left( ie^{i \theta} - \frac{1}{ie^{i \theta }}\right)} = e^{\frac{t}{2}\left( \omega - \frac{1}{\omega}\right)}. $$ This is the generating function for Bessel's functions, so $$ e^{\frac{t}{2}\left( \omega - \frac{1}{\omega}\right)} = \sum_{m = -\infty}^{\infty}J_{m}(kr)\omega^{m} = \sum_{m = -\infty}^{\infty}J_{m}(kr)i^{m}e^{im\theta}. \qquad (.1) $$ It may be represented as $$ \sum_{m = -\infty}^{\infty}J_{m}(kr)i^{m}e^{im\theta} = \sum_{m = -\infty}^{\infty}\sum_{n = 0}^{\infty}J_{m}(kr)i^{m}a_{mn}P_{n}(cos(\theta)). \qquad (.2) $$ I read that full expansion of plane wave expression into special functions polynomial is $$ e^{i(\mathbf k \cdot \mathbf r )} = \sum_{m = 0}^{\infty}(2m + 1)J_{m}(kr)i^{m}P_{m}(cos(\theta )). \qquad (.3) $$ So how to get $(.3)$ from $(.2)$ (or from $(.1)$)? I don't know what to do with $e^{im \theta}$.

Answer 1

Formula (.3) is not correct. Instead of $J_m(kr)$ it should be $j_m(kr)$ which correspond to Spherical Bessel functions. Work backwards from the spherical harmonic expansion

$$e^{i\mathbb{k}\cdot\mathbb{r}}=4\pi\sum_{\ell=0}^\infty i^\ell j_\ell(k r)\sum_{m=-k}^k Y^*_{\ell m}(\theta,\phi)Y_{\ell m}(\theta',\phi')$$

and use the spherical addition theorem specialized to the Legendre polynomial with $\cos(\gamma)=\cos(\theta)\cos(\theta')+\sin(\theta)\sin(\theta')\cos(\phi-\phi')$:

$$P_\ell(\cos\gamma) = \frac{4\pi}{2\ell +1}\sum_{m=-\ell}^\ell Y^*_{\ell m}(\theta,\phi)Y_{\ell m}(\theta',\phi')$$