Asymptotic expansion of Mathieu-Bessel series. II
We consider the asymptotic expansion of the Mathieu-Bessel series \[S_{\nu,\gamma}^{\mu}(a,b)=\sum_{n=1}^\infty \frac{n^\gamma K_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu>0, \nu\geq 0, b>0, \gamma\in {\bf R})\] as $|a|\to\infty$ in $|\arg\,a|<\pi/2$ with the other parameters held fixed, where $K_\nu(x)$ is the modified Bessel function of the second kind of order $\nu$. We employ a Mellin transform approach to determine an integral representation for $S_{\nu,\gamma}^{\mu}(a,b)$ involving the Riemann zeta function. Asymptotic evaluation of this integral involves appropriate residue calculations. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained. The expansion of the alternating variant of $S_{\nu,\gamma}^\mu(a,b)$ is also considered.