On a Multidimensional Volkenborn Integral and Higher Order Bernoulli Numbers

In particular, the values at x = 0 are called Bernoulli numbers of order k, that is, Bn (0) = Bn k) (see [1, 2, 4, 5, 9, 10, 14]). When k = 1, the polynomials or numbers are called ordinary. The polynomials Bn (x) and numbers Bn were first defined and studied by Norlund [9]. Also Carlitz [2] and others investigated their properties. Recently they have been studied by Adelberg [1], Howard [5], and Young [14]. In [l], Adelberg has given congruences for Bn which extended the Kummer congruences and has deduced information concerning the irreducibility of certain Bernoulli polynomials with order divisible by p. Howard [5] investigated other numbers related to the higher order Bernoulli numbers. Young [14] considered the p-adic integrals and measures to obtain congruences for the higher order Bernoulli numbers and polynomials. In this paper, using a multidimensional Volkenborn integral, we give a p-adic expression for Bernoulli number of order k. As an easy corollary we see that a p-adic expression for the higher order Bernoulli number is related to the sums of products of the ordinary Bernoulli numbers in Dilcher [4]. We give some examples. Our approach essentially coincides with the p-adic expression for the ordinary Bernoulli numbers