A CLASS OF IDENTITIES ASSOCIATED WITH DIRICHLET SERIES SATISFYING HECKE’S FUNCTIONAL EQUATION

We consider two sequences a(n) and b(n), 1 ≤ n < ∞, generated by Dirichlet series of the forms ∞ ∞ ∑ a(n) and <inf>n</inf> ∑ <inf>=1</inf> b <inf>μ</inf> (n <inf>sn</inf> ) λs n n=1 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<inf>ν</inf>, 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) = τ(n) and a(n) = r<inf>k</inf>(n), are examined, where τ(n) is Ramanujan’s arithmetical function and r<inf>k</inf>(n) denotes the number of representations of n as a sum of k squares. All but one of the examples appear to be new.