Closed expressions for some infinite series of Bessel and Struve functions (Q1089811)

The authors find closed expressions for four types of infinite series. Two of these expressions involve a Bessel function \(J_{\nu}(x)\) of the first kind. The other two involve Struve's function \(H_{\nu}(x)\). The results are of the following type \[ (1)\quad s_{- q}(e,x,\nu)=\sum^{\infty}_{k=1}\frac{\epsilon_ k}{k^ q(k^ 2- c^ 2)}\phi_{\nu}(kx) \] when \(q=2,4,6,\epsilon_ k=(-1)^{k-1}\), \(\phi_{\nu}(x)=x^{-\nu}J_{\nu}(x)\), (1) recovers the closed expression conjectured by H. Fettis in an unsolved problem [A summation of Bessel function, (Problem \(84-18^*)\), SIAM Rev. 26, 430-431 (1984)]. The Fettis problem has been already solved by \textit{E. Hansen} [SIAM Rev. 27, 445-457 (1985)] with a different method.