We consider two sequences a(n) and b(n), 1 ≤ n < ∞, generated by Dirichlet series of the forms ∞ ∞ ∑ a(n) and n ∑ =1 b μ (n sn ) 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 Kν, 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) = rk(n), are examined, where τ(n) is Ramanujan’s arithmetical function and rk(n) denotes the number of representations of n as a sum of k squares. All but one of the examples appear to be new.
- Bessel functions
- Classical arithmetic functions
- Functional equations
ASJC Scopus subject areas
- Applied Mathematics