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Given the Newtonian gravitational potential,

$$ \phi(\mathbf{x}) = - \int \frac{G \rho(\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|}$$

One can construct a 'multipole expansion' by using the Taylor series expansion of $1/|\mathbf{x} - \mathbf{x'}|$ about $\mathbf{x'} = 0$. The integral can then be written as,

$$ \phi(\mathbf{x}) = - \frac{GM}{r} - \frac{G}{r^3} x^k D^k - \frac{G}{2} Q^{kl} \frac{x^k x^l}{r^5} + ...$$

where summation over repeated indices is implied. $D^k$ is identified as the mass dipole and $Q^{kl}$ the mass quadrupole.

My question is how does this relate to the language of multipole moments, spherical harmonics, $m$ and $l$ numbers etc? Why should we describe $D^k$ and $Q^{kl}$ and a 'dipole' and 'quadrupole'? All I can see so far is a Taylor expansion? How does this 'translate' into the language of angles and multipole moments as outlined in e.g. this Wikipedia entry?

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It is not a complete answer, but it is something. Recall the generating function for Legendre polynomials are: $$ \frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n=0}^\infty P_n(x)t^n $$

Therefore: $$ \begin{align} \frac{1}{|\mathbf x - \mathbf x'|} &= \frac{1}{\sqrt{x^2 + x'^2 - 2xx'\cos(\mathbf x, \mathbf x')}}\\ &= \frac{1}{x}\frac{1}{\sqrt{1 + \left(\frac{x'}{x}\right)^2 - \frac{2x'}{x}\cos(\mathbf x, \mathbf x')}}\\ &= \frac{1}{x}\sum_{n=0}^\infty P_n(\cos(\mathbf x, \mathbf x'))\left(\frac{x'}{x}\right)^n \end{align} $$

This is how the multipole expansion relates to the Legendre polynomials. Substituting this at the potential, you will have integral forms over spherical coordinates of the multipole moments of the distribution $\rho(\mathbf x')$.

To actually carry out integration, you will have to convert $\cos(\mathbf x, \mathbf x')$ into the actual angles $\theta, \phi$ of the spherical coordinate system. Doing this, yields the spherical harmonics you were looking for.

Perhaps this wikipedia page is what you are looking for.

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