A Class of Identities Associated with Dirichlet Series Satisfying Hecke's Functional Equation

Abstract

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ 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 $K_ν$, 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 $a(n)=r_k(n)$, are examined, where $τ(n)$ is Ramanujan's arithmetical function and $r_k(n)$ denotes the number of representations of $n$ as a sum of $k$ squares. Most of the six special cases appear to be new.