New sum rule for products of Bessel functions with application to plasma physics

In our investigations of the linear theory of the stability of relativistic beam-plasma systems immersed in a magnetic field we have been led to consider sum rules for an infinite series of products of Bessel functions of the form 𝒥∞n = −∞(n jJ2n)/ (n+μ). In this work we report on the sum of this series treated as a special case of a more general infinite series. We also mention the extension of the results beyond the range of the parameters for which formulae are explicitly given and indicate how intermediate results obtained may be useful in their own right. Finally, an additional application of our result is indicated.