On Euler numbers, polynomials and related p-adic integrals

In this note we give a new proof of Witt's formula for Euler numbers, which are related to some known or new identities involving the Euler numbers. We also obtain a brief proof of a classical result on Euler numbers modulo of two due to M.A. Stern using the approach of p-adic integration, which was recently proved by G. Liu, and Z.-W. Sun. Finally some explicit formulas for Genocchi numbers are proved and applications are given.