Convolution identities and lacunary recurrences for Bernoulli numbers Original Research Article

We extend Eulerʹs well-known quadratic recurrence relation for Bernoulli numbers, which can be written in symbolic notation as (B0+B0)n=−nBn−1−(n−1)Bn, to obtain explicit expressions for (Bk+Bm)n with arbitrary fixed integers k,mgreater-or-equal, slanted0. The proof uses convolution identities for Stirling numbers of the second kind and for sums of powers of integers, both involving Bernoulli numbers. As consequences we obtain new types of quadratic recurrence relations, one of which gives B6k depending only on B2k,B2k+2,…,B4k.