Convolution identities and lacunary recurrences for Bernoulli numbers

Abstract

We extend Euler's well-known quadratic recurrence relation for Bernoulli numbers, which can be written in symbolic notation as ( B 0 + B 0 ) n = − n B n − 1 − ( n − 1 ) B n , to obtain explicit expressions for ( B k + B m ) n with arbitrary fixed integers k , m ⩾ 0 . 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 B 6 k depending only on B 2 k , B 2 k + 2 , … , B 4 k .