Two general series identities involving modified Bessel functions and a class of arithmetical functions

We consider two sequences a(n) and, b(n), 1 ≤ n < ∞ generated by Dirichlet series {equation presented} satisfying a familiar functional equation involving the gamma function Γ(s). Two general identities are established. The first involves the modified Bessel function K_μ(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_μ(z), the Bessel functions of imaginary argument I_μ(z), and ordinary hypergeometric functions ₂F₁(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 τ(n), the number of representations of n as a sum of k squares rk(n), and primitive Dirichlet characters Χ(n).