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Graf's addition formula for Bessel functions states that: $$ H_{n}^{(1)}\left(\sqrt{a^2+b^2-2ab\cos(\theta)}\right)=\sum_{m=-\infty}^{\infty}H_{m+n}^{(1)}(a)J_{m}(b)e^{im\theta} $$ for $a>b>0$. I would like to know if a similar formula exists for spherical Bessel functions i.e. Bessel functions of half integer order i.e. something like $$ h_{n}^{(1)}\left(\sqrt{a^2+b^2-2ab\cos(\theta)}\right)=\sum_{m=-\infty}^{\infty}h_{m+n}^{(1)}(a)j_{m}(b)e^{im\theta} $$ I can't find anything in the literature, but I would like to know for sure that one doesn't exist.

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Graf's addition theorem (9.1.79 here) $$ H_{\nu}^{(1)}\left(\sqrt{a^2+b^2-2ab \cos(\theta)}\right)=\sum_{m=-\infty}^{\infty}H_{m+\nu}^{(1)}(a)J_{m}(b)e^{im\theta}$$ holds for arbitrary $\nu$. Setting $\nu = n + \frac12$, we obtain the addition theorem $$h_{n}^{(1)}\left(\sqrt{a^2+b^2-2ab \cos(\theta)}\right)=\sum_{m=-\infty}^{\infty}h_{m+n}^{(1)}(a)J_{m}(b)e^{im\theta} $$ for the spherical Bessel functions. Note that $J_m$, different from your proposal, is a conventional Bessel function.

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