NEW FORMULAS FOR THE BERNOULLI AND EULER POLYNOMIALS AT RATIONAL ARGUMENTS

We prove theorems on the values of the Bernoulli polynomials B-n(x) with n = 2, 3, 4,..., and the Euler polynomials E(n)(x) with n = 1, 2, 3,... for 0 less than or equal to x less than or equal to 1 where x is rational. B-n(x) and E(n)(x) are expressible in terms of a finite combination of trigonometric functions and the Hurwitz zeta function zeta(z, alpha). The well-known argument-addition formulae and reflection property of B-n(x) and E(n)(x), extend this result to any rational argument. We also deduce new relations concerning the finite sums of the Hurwitz zeta function and sum some classical trigonometric series.