Universal Higher Order Bernoulli Numbers and Kummer and Related Congruences

Abstract

We define higher or arbitrary order universal Bernoulli numbers and higher order universal Bernoulli–Hurwitz numbers. We deduce a universal first-order Kummer congruence and a congruence for the higher order universal Bernoulli–Hurwitz numbers from Clarke's universal von Staudt theorem. We also establish other Kummer-type congruences for the higher order universal Bernoulli numbers and for the universal Nörlund polynomials, generalizing the author's previous work.