Computation of Euler's type sums of the products of Bernoulli numbers

Abstract

In this work, the authors present several formulas which compute the following Euler's type and Dilcher's type sums of the products of Bernoulli numbers Bn: Ωn(m) {colon equals} under(∑, frac(j1 + ⋯ + jm = n, (j1, ..., jm ≧ 1))) fenced(frac(2 n, 2 j1, ..., 2 jm)) B2 j1 ⋯ B2 jm and Δn(m) {colon equals} under(∑, frac(j1 + ⋯ + jm = n, (j1, ..., jm ≧ 0))) fenced(frac(2 n, 2 j1, ..., 2 jm)) B2 j1 ⋯ B2 jm respectively, where fenced(frac(n, k1, ..., km)) = frac(n !, k1 ! ⋯ km !) denotes, as usual, the multinomial coefficient. © 2008 Elsevier Ltd. All rights reserved.