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

Abstract

We consider two sequences 𝑎¹𝑛ºand𝑏¹𝑛º, 𝑛1, generated by Dirichlet series ∑︁𝑛=1𝑎¹𝑛º𝜆𝑠𝑛 and ∑︁𝑛=1𝑏¹𝑛º𝜇𝑠𝑛 satisfying a familiar functional equation involving the gamma function Γ¹𝑠º. Two general identities are established. The first involves the modified Bessel function 𝐾𝜇¹𝑧º 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 𝐾𝜇¹𝑧º, the Bessel functions of imaginary argument 𝐼𝜇¹𝑧º, and ordinary hypergeometric functions 2𝐹¹𝑎 𝑏;𝑐;𝑧º. 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 𝜏¹𝑛º the number of representations of𝑛as a sum of 𝑘squares𝑟𝑘¹𝑛º and primitive Dirichlet characters 𝜒¹𝑛º.