  We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by
Dirichlet series of the forms
  $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad
  \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ 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. Seven
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. Most of the six special
cases appear to be new.
