A class of identities associated with Dirichlet series satisfying Hecke's functional equation

Abstract

We consider two sequences a(n) and b(n), 1≤n<∞, generated by Dirichlet series of the forms ∑n=1∞a(n)λsnand∑n=1∞b(n)μsn, satisfying a familiar functional equation involving the gamma function Γ(s). A general identity is established. Appearing on one side is an infinite series involving a(n) and modified Bessel functions Kν, wherein on the other side is an infinite series involving b(n) that is an analogue of the Hurwitz zeta function. Seven special cases, including a(n)=τ(n) and a(n)=rk(n), are examined, where τ(n) is Ramanujan's arithmetical function and rk(n) denotes the number of representations of n as a sum of k squares. Most of the six special cases appear to be new.