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

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Abstract

We show that the Ramanujan sequence (θ n ) 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 (θ n ) 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.

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Authors and Affiliations

  1. Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009, Zaragoza, Spain

    José A. Adell & P. Jodrá

Authors
  1. José A. Adell
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  2. P. Jodrá
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Correspondence to José A. Adell.

Additional information

This work was supported by research projects BFM2002-04163-C02-01 and DGA E-12/25, and by FEDER funds.

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Adell, J.A., Jodrá, P. On the complete monotonicity of a Ramanujan sequence connected with e n . Ramanujan J 16, 1–5 (2008). https://doi.org/10.1007/s11139-007-9088-7

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  • DOI: https://doi.org/10.1007/s11139-007-9088-7

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