An estimate for the powers of the generating function of the Bernoulli numbers

Abstract

Let t ≥ 0 be real and let e (t) denote the integal , which is the integral analogue to the power series of the exponential function exp(t). Then, by using the Laplace transform Δ(x) of the function δ(t):= exp(t) − e (t) for all real t ≠ 0, we will get the estimate 0 < B n (t) − PP 0[B n ](t) < 1, which is sharp at t = ±∞, where B n is defined by B n (t):= (1 − exp(−t))−n and PP 0[B n ] denotes the principal part of the Laurent expansion of B n at t = 0. We will see that the expression between the two ‘<’ signs strictly increases (on R) and will express its limit at t = 0 in terms of the ‘generalized Bernoulli numbers’ , i.e. by the coefficients of the Taylor expansion of [z/(exp(z) − 1)] n at z = 0.