Congruences involving Bernoulli polynomials Original Research Article

Let image be the Bernoulli polynomials. In the paper we establish some congruences for image, where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for image and the sum image, where image is the class number of the quadratic field image of discriminant d and p-regular functions are those functions f such that image are rational p-integers and image for image . We also establish many congruences for Euler numbers.