=+Check the defining conditions (A.10) of the Bernoulli polynomials by evaluating limt→0 f(t, x),  1 0 f(t, x) dx, and ∂

∂x f(t, x). Hence, the coefficients Bn(x) are the Bernoulli polynomials. The special case f(t, 0) = t et − 1 = ∞

n=0 Bn n!

t n (12.16)

gives the exponential generating function of the Bernoulli numbers.