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On a function studied by Ramanujan and connected with discrete mixtures of gamma densities

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Abstract

We consider discrete mixtures of gamma densities with appropriate parameters, and characterize the complete monotonicity on any interval (c,∞) (c≥0) in terms of its defining parameters. This extends a previous result of Boas, after observing that the function

$$\phi(t)=\sum_{k=1}^{\infty}\frac{k^{k-1}}{k!}t^{k-1}e^{-kt}\quad (t\geq0)$$

is, up to a constant, a special case of such mixtures. Moreover, we study convexity and subadditivity properties of φ λ (λ∈R) and we present several functional inequalities involving φ.

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Authors and Affiliations

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

    José A. Adell

  2. Morsbacher Str. 10, 51545, Waldbröl, Germany

    Horst Alzer

Authors
  1. José A. Adell
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  2. Horst Alzer
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Correspondence to Horst Alzer.

Additional information

This work has been partially supported by research grants MTM2008-06281-C02-01/MTM and DGA E-64, and by FEDER funds.

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Adell, J.A., Alzer, H. On a function studied by Ramanujan and connected with discrete mixtures of gamma densities. Ramanujan J 20, 127–151 (2009). https://doi.org/10.1007/s11139-009-9192-y

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  • DOI: https://doi.org/10.1007/s11139-009-9192-y

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