## Abstract

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 K_{ν}, 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_{k}(n), are examined, where τ(n) is Ramanujan’s arithmetical function and r_{k}(n) denotes the number of representations of n as a sum of k squares. All but one of the examples appear to be new.

Original language | English (US) |
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Pages (from-to) | 4785-4799 |

Number of pages | 15 |

Journal | Proceedings of the American Mathematical Society |

Volume | 150 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2022 |

## Keywords

- Bessel functions
- Classical arithmetic functions
- Functional equations

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics