Summing Sneddon–Bessel series explicitly

We sum in a closed form the Sneddon–Bessel series where is an integer, with and are the zeros of the Bessel function of order . In most cases, the explicit expressions for these sums involve hypergeometric functions . As an application, we prove some extensions of the Kneser–Sommerfeld expansion. For instance, we show that if and (here, denotes the Bessel function of the second kind), which becomes the Kneser–Sommerfeld expansion when .