Raabe's Identity and Covering Systems
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
is an MA-covering function for a system of arithmetic
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,
(Concerned with sequences
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.
Journal of Integer Sequences home page