A Bernstein function related to Ramanujan's approximations of exp(n)

Abstract

Ramanujan's sequence θ(n),n=0,1,2,..., is defined by en/2 =∑j=0 n-1nj/j!+nn/n! θ(n). It is possible to define, in a simple manner, the function θ(x) for all nonnegative real numbers x. We show that the function λ(x):=x (θ(x)-1/3) is a Bernstein function on [0,∞), that is, λ(x) is nonnegative with completely monotonic derivative on [0,∞). This implies some earlier results concerning complete monotonicity of the function θ(x) on [0,∞). © 2013 Springer Science+Business Media, LLC.