Expanding potential in Legendre polynomials (or spherical harmonics)

[rnielsen25]

Homework Statement

Expand $$\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }$$ in legendre polynomials.

Relevant Equations

$$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$

Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid \vec{r}-\vec{r}^{\prime}\right]}=\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2 r r^{\prime}\left(\hat{r} \cdot \hat{r}^{\prime}\right)}}= \sum_{\ell=0}^{\infty} \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}} P\left(\hat{r} \cdot \hat{r}^{\prime}\right) \\ &=\sum_{\ell=0}^{\infty} \frac{4 \pi}{2 \ell+1} \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}} \sum_{m=-\ell}^{\ell} Y_{\ell m}^{\star}\left(\vartheta^{\prime}, \varphi^{\prime}\right) Y_{\ell m}(\vartheta, \varphi) \end{aligned} $$ Where $r_{<}$ and $r_{>}$ represent the smaller and larger of $r$ and $r^{\prime}$.

But I need to expand
$$\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }=\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2 r r^{\prime}\left(\hat{r} \cdot \hat{r}^{\prime}\right)+a}}$$
in a similar way. However, I can't seem to pull out a factor of $r$ or $r'$ to get the generating function as you can above, because of the addition of $a$.
So how do I expand this expression in legendre polynomials?

 

[TSny]

But I need to expand
$$\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }=\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2 r r^{\prime}\left(\hat{r} \cdot \hat{r}^{\prime}\right)+a}}$$
in a similar way. However, I can't seem to pull out a factor of $r$ or $r'$ to get the generating function as you can above, because of the addition of $a$.

You might try pulling out a factor of $\sqrt{r^2+a}$ : $$\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2 r r^{\prime}\left(\hat{r} \cdot \hat{r}^{\prime}\right)+a}} = \frac{1}{\sqrt{r^2+a}} \frac{1}{\sqrt{1 -2xt+t^2}}$$ where you will need to determine expressions for $t$ and $x$.