Journal of Integer Sequences, Vol. 23 (2020), Article 20.2.8

Raabe's Identity and Covering Systems

Yevgenya Movshovich
University of Illinois at Urbana-Champaign
Urbana, IL 61801


A connection between covering systems and Bernoulli polynomials established by Fraenkel, Beebee, and Porubsky asserts that a function Σi=1q μiχAi(k) is an MA-covering function for a system of arithmetic sequences {Ai}i=1q if and only if it satisfies a generalized Raabe identity. The connection is used to derive recurrence formulas for the Bernoulli numbers. Here we show that the generalized Raabe identity is a sum of Raabe’s identities for Ai multiplied by μi, and this sum is the direct source of the above recurrence formulas. We show that many of these formulas are special cases of the original multiplication formula of Raabe. We find new applications of the connection to the covering systems.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A027641 A027642.)

Received March 28 2019; revised versions received April 2 2019; January 31 2020; February 5 2020; February 11 2020. Published in Journal of Integer Sequences, February 20 2020.

Return to Journal of Integer Sequences home page