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

Abstract

We consider two sequences a(n) and b(n), 1≤n<∞, generated by Dirichlet series ∑n=1∞a(n)λsnand∑n=1∞b(n)μsn,satisfying a familiar functional equation involving the gamma function Γ(s). Two general identities are established. The first involves the modified Bessel function Kμ(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 Kμ(z), the Bessel functions of imaginary argument Iμ(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).