On some infinite series involving the zeros of Bessel functions of the first kind

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γ v, n ( n = l, 2, 3,…) the positive zeros of J v (z), then, in certain physical applications, we are interested in finding the values of the sums and where m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S 2m,v which enables the value of any sum to be calculated as a rational function of the order v of the Bessel function. Similar results are given in § 5 for the sum T 2m,v .