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Series representation of multiplication of two Bessel function $J_{\mu}(az) J_{\nu}(bz)$ is in terms of sum of hypergeometric functions $_2F_1$, it given in book Treatise on Theory of Bessel Functions by Watson.

What is representation of multiplication of two modified Bessel function of different kinds

$I_{\mu}(az) K_{\nu}(bz) = ?$

Interesting to me is case of $\nu$ is an integer $n$.

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    $\begingroup$ $K_\nu$ is a linear sum of $I_\nu$ and $I_{-\nu}$, unless $\nu$ is integer. Since $I_\nu(x)/J_\nu(ix)=const$, this should be easy. $\endgroup$
    – Nemo
    Commented Jul 21, 2017 at 16:17
  • $\begingroup$ @Nemo What if $\nu$ is an integer? $\endgroup$
    – Nigel1
    Commented Jul 23, 2017 at 13:18
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    $\begingroup$ I don't know, but I guess one will obtain a derivative of hypergeometric function with respect to a parameter, which will be a series involving generalized Harmonic numbers. $\endgroup$
    – Nemo
    Commented Jul 23, 2017 at 15:05

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