We consider two sequences a(n) and b(n), 1\leq n<\infty, generated by Dirichlet series of the forms \begin{equation*} \sum _{n=1}^{\infty }\dfrac {a(n)}{\lambda _n^{s}}\qquad \text {and}\qquad \sum _{n=1}^{\infty }\dfrac {b(n)}{\mu _n^{s}}, \end{equation*} satisfying a familiar functional equation involving the gamma function \Gamma (s). A general identity is established. Appearing on one side is an infinite series involving a(n) and modified Bessel functions K_{\nu }, wherein on the other side is an infinite series involving b(n) that is an analogue of the Hurwitz zeta function. Six special cases, including a(n)=\tau (n) and a(n)=r_k(n), are examined, where \tau (n) is Ramanujan's arithmetical function and r_k(n) denotes the number of representations of n as a sum of k squares. All but one of the examples appear to be new. [ABSTRACT FROM AUTHOR]