We consider two sequences
a
(
n
)
a(n)
and
b
(
n
)
b(n)
,
1
≤
n
>
∞
1\leq n>\infty
, generated by Dirichlet series of the forms
∑
n
=
1
∞
a
(
n
)
λ
n
s
and
∑
n
=
1
∞
b
(
n
)
μ
n
s
,
\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
Γ
(
s
)
\Gamma (s)
. A general identity is established. Appearing on one side is an infinite series involving
a
(
n
)
a(n)
and modified Bessel functions
K
ν
K_{\nu }
, wherein on the other side is an infinite series involving
b
(
n
)
b(n)
that is an analogue of the Hurwitz zeta function. Six special cases, including
a
(
n
)
=
τ
(
n
)
a(n)=\tau (n)
and
a
(
n
)
=
r
k
(
n
)
a(n)=r_k(n)
, are examined, where
τ
(
n
)
\tau (n)
is Ramanujan’s arithmetical function and
r
k
(
n
)
r_k(n)
denotes the number of representations of
n
n
as a sum of
k
k
squares. All but one of the examples appear to be new.