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 \begin{equation*} \sum _{n=1}^{\infty }\dfrac {a(n)}{\lambda _n^{s}}\qquad \text {and}\qquad \sum _{n=1}^{\infty }\dfrac {b(n)}{\mu _n^{s}}, \end{equation*} satisfying a familiar functional equation involving the gamma function $\Gamma (s)$. A general identity is established. Appearing on one side is an infinite series involving $a(n)$ and modified Bessel functions $K_{\nu }$, wherein on the other side is an infinite series involving $b(n)$ that is an analogue of the Hurwitz zeta function. Six special cases, including $a(n)=\tau (n)$ and $a(n)=r_k(n)$, are examined, where $\tau (n)$ is Ramanujan’s arithmetical function and $r_k(n)$ denotes the number of representations of $n$ as a sum of $k$ squares. All but one of the examples appear to be new.