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We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated byDirichlet 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 functionalequation involving the gamma function $\Gamma(s)$. Two general identities areestablished. The first involves the modified Bessel function $K_{\mu}(z)$, andcan be thought of as a 'modular' or 'theta' relation wherein modified Besselfunctions, instead of exponential functions, appear. Appearing in the secondidentity 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 identitiesare new. The arithmetical functions appearing in the identities includeRamanujan's arithmetical function $\tau(n)$; the number of representations of$n$ as a sum of $k$ squares $r_k(n)$; and primitive Dirichlet characters$\chi(n)$.

Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions