On the complete monotonicity of a Ramanujan sequence connected with e n

• DOI: 10.1007/s11139-007-9088-7
• Publication Date: 2008-05-10T14:31:34Z
• Authors: José A. Adell; P. Jodrá
• Keywords: 0101 mathematics; 01 natural sciences

We show that the Ramanujan sequence (θn)n≥0 defined as the solution to the equation $$\frac{e^{n}}{2}=\sum_{k=0}^{n-1}\frac{n^{k}}{k!}+\frac{n^{n}}{n!}\theta_{n}$$ is completely monotone. Our proof uses the fact that (θn)n≥0 coincides, up to translation and renorming, with the moment sequence of a probability distribution function on [0,1] involving the two real branches of the Lambert W function.