Graf's Addition Theorem

Graf's addition theorem for Bessel functions is given in Abramowitz and Stegun 1964. It is a special case of a general addition theorem called Neumann's addition theorem. Details can be found in Abramowitz and Stegun 1964 online. We express the theorem in the following form

$C_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} = \sum_{\mu = - \infty}^{\infty} C_{\nu + \mu} (\eta R_{jl}) \, J_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,$

where $C_\nu$ can represent any of the Bessel functions $\,\!J_\nu$, $\,\!I_\nu$, $\,\!Y_\nu$, $\,\!K_\nu$, $H_\nu^{(1)}$, and $H_\nu^{(2)}$, $(r_j,\theta_j)\,\!$ and $(r_l,\theta_l)\,\!$ are polar coordinates centred at two different positions with global coordinates $\boldsymbol{O}_j$, $\boldsymbol{O}_l$, and $(R_{jl},\vartheta_{jl})$ are the polar coordinates of $\boldsymbol{O}_l$ with respect to $\boldsymbol{O}_j$. This expression is valid only provided that $\,\!r_l < R_{jl}$ ( although this restriction is unnecessary if $\,\!C=J$ and $\,\!\nu$ is an integer).

Explicit versions of the theorem are given below,

$H_\nu^{(1)}(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} = \sum_{\mu = - \infty}^{\infty} H^{(1)}_{\nu + \mu} (\eta R_{jl}) \, J_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,$ $K_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} = \sum_{\mu = - \infty}^{\infty} K_{\nu + \mu} (\eta R_{jl}) \, I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,$

This theorem form the basis for Kagemoto and Yue Interaction Theory.