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How can one prove the following identity: $$ V_k(r_1, r_2) = {2k+1\over 2 r_1 r_2}\int_{|r_1 - r_2|}^{r_1+r_2} e^{-{r\over D}} P_k\left(r_1^2 - r^2 + r_2^2 \over 2 r_1 r_2 \right) d r= $$ $$ =(2k+1) {I_{k+{1\over2}}\left(r_<\over D\right) \over \sqrt{r_<}} {K_{k+{1\over2}}\left(r_>\over D\right) \over \sqrt{r_>}} $$ ? Here $r_<=\min(r_1, r_2)$, $r_>=\max(r_1, r_2)$, $I_\nu$ and $K_\nu$ are the modified Bessel functions of the first and second kind, $P_k(x)$ are Legendre polynomials and $k=0, 1, 2, 3, \dots$.

I verified the identity numerically, so I know that it works, but I didn't figure out how one could prove it. I know it has something to do with Gegenbauer's addition theorem and the equation 10.23.8 at [1]. The $V_k(r_1, r_2)$ is just the coefficient in the Legendre expansion of: $$ {e^{{|{\bf r}_1 - {\bf r}_2|}\over D}\over |{\bf r}_1 - {\bf r}_2|} =\sum_{k=0}^\infty V_k(r_1, r_2) P_k(\cos\theta) $$ Note: when $D\to\infty$, then $V_k$ becomes ${r_<^k \over r_>^{k+1}}$, which follows from the properties of the Bessel functions and this is just the well known multipole expansion [2].

[1] http://dlmf.nist.gov/10.23#E8

[2] http://en.wikipedia.org/wiki/Legendre_polynomials#Applications_of_Legendre_polynomials_in_physics

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Cross-listed to physics.stackexchange.com/q/41167/2451 –  Qmechanic Oct 18 '12 at 20:55

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