Higher-order recurrences for Bernoulli numbers

Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as n(B0+B0)=−nBn−1−(n−1)Bn, is extended to n(Bk1+⋯+Bkm) for m⩾2 and arbitrary fixed integers k1,…,km⩾0. In the general case we prove an existence theorem for Euler-type formulas, and for m=3 we obtain explicit expressions. This extends the authors' previous work for m=2.