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

TL;DR

This paper derives a general identity relating two sequences generated by Dirichlet series satisfying Hecke's functional equation, involving Bessel functions and zeta functions, with several new special cases examined.

Contribution

It introduces a new general identity connecting Dirichlet series with functional equations, Bessel functions, and zeta functions, including several novel special cases.

Findings

Established a general identity linking Dirichlet series and special functions.

Identified seven special cases, many of which are new.

Connected classical arithmetic functions with advanced analytical identities.

Abstract

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ 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. Seven 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. Most of the six special cases appear to be new.