Berndt, Bruce C.Bruce C.BerndtDixit, AtulAtulDixitGupta, RajatRajatGuptaZaharescu, AlexandruAlexandruZaharescu2025-08-282025-08-282022-04-01http://arxiv.org/abs/2204.09887https://d8.irins.org/handle/IITG2025/20103We consider two sequences a(n) and b(n), 1?n<?, generated by Dirichlet series ?n=1?a(n)?snand?n=1?b(n)?sn,satisfying a familiar functional equation involving the gamma function ?(s). Two general identities are established. The first involves the modified Bessel function K?(z), 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 K?(z), the Bessel functions of imaginary argument I?(z), and ordinary hypergeometric functions 2F1(a,b;c;z). 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 ?(n); the number of representations of n as a sum of k squares rk(n); and primitive Dirichlet characters ?(n).en-USBessel functionsFunctional equationsClassical arithmetic functionsNumber Theorye-Printe-Print123456789/555