The evaluation of single Bessel function sums

We examine convergent representations for the sums of Bessel functions X∞ n=1 Jν(nx) nα (x > 0) and X∞ n=1 Kν(nz) nα (<(z) > 0), together with their alternating versions, by a Mellin transform approach. We take α to be a real parameter with ν > − 1 2 for the first sum and ν ≥ 0 for the second sum. Such representations enable easy computation of the series in the limit x or z → 0+. Particular attention is given to logarithmic cases that occur for certain values of α and ν