Generalized Volkenborn Integrals Associated with p -Adic Distributions and the Bernoulli Numbers

Abstract: Our goal is to give a formula representing the Bernoulli numbers by p-adic distributions. We consider p-adic distributions on the ring of p-adic integers which are invariant by rotations around the origin, and define a generalization of the Vokenborn integrals with respect to such distributions. It is shown the generalized Volkenborn integrals of power functions, and of negative powers of the p-adic norm converge under some conditions on the distributions, and their universal relation to the Bernoulli numbers is presented.