A connection between covering systems and Bernoulli polynomials established by Fraenkel, Beebee, and Porubsky asserts that a function Σ_i=1^q μ_iχ_Ai(k) is an M_A-covering function for a system of arithmetic sequences {A_i}_i=1^q 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 A_i multiplied by μ_i, 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.