A Short Proof of the Explicit Formula for Bernoulli Numbers

As Gould pointed out in his survey article [2], formula (1) appeared for the first time in [3]. Garabedian [1] rediscovered (1) and proved it by using Cesaro sums. We introduce the set {a,,k } of integers given by k-i k an,k (-1) (-)j (j + ?1)n j=o for n = 0, 1, 2, ... and k = 1, 2, 3, .... The numbers an,k can be expressed in terms of finite differences. By the mth difference of a function g(x) we mean the quantity m (-) Amg(x) O'= (-1)m-jg(x + j) (m 0, 1,2,...), j=O()