  We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by
Dirichlet series
  $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad
  \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ satisfying a familiar functional
equation involving the gamma function $\Gamma(s)$. Two general identities are
established. The first involves the modified Bessel function $K_{\mu}(z)$, and
can be thought of as a 'modular' or 'theta' relation wherein modified Bessel
functions, instead of exponential functions, appear. Appearing in the second
identity are $K_{\mu}(z)$, the Bessel functions of imaginary argument
$I_{\mu}(z)$, and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$.
Although certain special cases appear in the literature, the general identities
are new. The arithmetical functions appearing in the identities include
Ramanujan's arithmetical function $\tau(n)$; the number of representations of
$n$ as a sum of $k$ squares $r_k(n)$; and primitive Dirichlet characters
$\chi(n)$.
