A note on a modified Bessel function integral

We investigate the integral $\int_0^\infty \cosh^μ\!t\,K_ν(z\cosh t)\,dt \qquad \Re(z)>0,$ where $K$ denotes the modified Bessel function, for non-negative integer values of the parameters $μ$ and $ν$. When the integers are of different parity, closed-form expressions are obtained in terms of $z^{-1}e^{-z}$ multiplied by a polynomial in $z^{-1}$ of degree dependent on the sign of $μ-ν$.

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