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By experimenting in Mathematica, I have found the following expression for the integral: $$ \int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}^{m}\left(\frac{\sigma^{2}-a^{2}-b^{2}}{2ab}\right)d\sigma=\\ aj_{n'}(a)\sum_{k=0}^{\min(n,n')-m}c_{n,n'}^{m,k}\frac{h_{n+n'-m-k}^{(1)}(b)}{b^{k+m-1}} $$

where $$ c_{n,n'}^{m,k}=\frac{(-1)^{k+m+n'}(n+m)!(n'+m)!(2k+2m)!}{2^{k+m-1}k!(k+m)!(k+2m)!(n-m-k)!(n'-m-k)!} $$ and $j_{n}$ are the spherical Bessel functions of the first kind, $h_{n}^{(1)}$ are the spherical Hankel functions of the first kind, $P_{n}^{m}$ are the associated Legendre polynomials, $b>a>0$ and $m,n,n'$ are non-negative integers such that $m\leq \min(n,n')$.

An equivalent, potentially tidier, formulation is: $$ \lim_{\delta\to 0}\frac{1}{\delta^{n'}}\int_{-1}^{1}h_{n}^{(1)}(b(1+\delta\sigma))P_{n}^{m}\left(1-\frac{\delta^{2}(1-\sigma^{2})}{2(1+\delta\sigma)}\right)P_{n'}^{m}\left(\sigma-\frac{\delta(1-\sigma^{2})}{2}\right)(1+\delta\sigma)d\sigma=\frac{2^{n'}(n')!}{(2n'+1)!}\sum_{k=0}^{\min(n,n')-m}c_{n,n'}^{m,k}\frac{h_{n+n'-m-k}^{(1)}(b)}{b^{k+m-n'}} $$ The result can be seen by running the code:

j[n_?IntegerQ, x_] := 
  Simplify[(-x)^n Nest[D[#, z]/z &, Sin[z]/z, n] /. z -> x];
h[n_?IntegerQ, x_] := 
  Simplify[-I (-x)^n Nest[D[#, z]/z &, Exp[I z]/z, n] /. z -> x];
m = 0;
n = 2;
np = 1;
int = Assuming[b > a > 0, 
  FullSimplify[
   Integrate[\[Sigma] SphericalHankelH1[n, \[Sigma]] LegendreP[n, 
      m, (\[Sigma]^2 - a^2 + b^2)/(2 b \[Sigma])] LegendreP[np, 
      m, (\[Sigma]^2 - a^2 - b^2)/(2 a b)], {\[Sigma], b - a, b + a}]]]
SolveAlways[
  int Exp[-I b]/(a j[np, a]) == 
   Sum[a1[k] h[n + np - k - m, b] Exp[-I b]/b^(k + m - 1), {k, 0, 
     Min[n, np] - m}], b][[1, All, 2]]

where any combination of $m,n,n'$ such that $m\leq \min(n,n')$ can be chosen. The first output gives the full expression for the integral; the second output shows the coefficients, $c_{n,n'}^{m,k}$.

The integral arises as the result of considering the scattering from multiple spheres. Essentially modes on a sphere can be represented as 'vector spherical harmonics'. These can be written in terms of spherical Bessel functions and Legendre polynomials. What the integral represents is the interaction between charge distributions on different spheres described by these modes.

I would like to be able to formally show this but (unsurprisingly) I'm having a few difficulties and I haven't been able to find what I'm looking for in places like DLMF and Gradshteyn and Ryzhik. I've tried using various definitions and expansions for the Legendre and spherical Hankel functions but it all just gets too messy. In the second formulation, I've tried expanding out the Legendre functions in powers of $\delta$, but, again, it gets very messy. An inductive proof would be elegant, but I don't see how to do this either.

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  • 3
    $\begingroup$ How the hell did you find that summation formula?! $\endgroup$
    – K.defaoite
    Oct 8, 2021 at 10:24
  • $\begingroup$ Mathematica was able to the integrals analytically for individual integer values of m,n and n'. From that I was (eventually) able to work out the pattern for general integers, and check it by numerical integration. $\endgroup$
    – Chris
    Oct 8, 2021 at 11:17
  • $\begingroup$ Would the summation be finite because of the minimum operator? $\endgroup$ Oct 11, 2021 at 12:36
  • $\begingroup$ You could take the summation up to infinity, provided you define the factorial of negative integers as infinity. $\endgroup$
    – Chris
    Oct 11, 2021 at 12:52
  • 1
    $\begingroup$ If the current bounty doesn't yield anything, I'd love to see this posted to Mathoverflow to see if someone there can answer. $\endgroup$
    – Yly
    Jun 21, 2022 at 4:08

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