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On a Ramanujan equation connected with the median of the gamma distribution


Authors: J. A. Adell and P. Jodrá
Journal: Trans. Amer. Math. Soc. 360 (2008), 3631-3644
MSC (2000): Primary 41A60; Secondary 60E05
DOI: https://doi.org/10.1090/S0002-9947-07-04411-X
Published electronically: December 20, 2007
MathSciNet review: 2386240
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider the sequence $ (\theta_n)_{n\ge 0}$ solving the Ramanujan equation

$\displaystyle \frac{e^n}{2}=\sum_{k=0}^{n}\frac{n^k}{k!}+\frac{n^n}{n!}\,(\theta_n-1),\qquad n=0,1,\dots. $

The three main achievements are the following. We introduce a continuous-time extension $ \theta(t)$ of $ \theta_n$ and show its close connections with the medians $ \lambda_n$ of the $ \Gamma(n+1,1)$ distributions and the Charlier polynomials. We give upper and lower bounds for both $ \theta(t)$ and $ \lambda_n$, in particular for $ \theta_n$, which are sharper than other known estimates. Finally, we show (and at the same time complete) two conjectures by Chen and Rubin referring to the sequence of medians $ (\lambda_n)_{n\ge 1}$.


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Additional Information

J. A. Adell
Affiliation: Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: adell@unizar.es

P. Jodrá
Affiliation: Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: pjodra@unizar.es

DOI: https://doi.org/10.1090/S0002-9947-07-04411-X
Keywords: Central limit theorem, Charlier polynomials, forward difference, gamma distribution, median, Poisson process, Ramanujan's equation
Received by editor(s): November 27, 2005
Received by editor(s) in revised form: April 27, 2006
Published electronically: December 20, 2007
Additional Notes: This work was supported by research projects BFM2002-04163-C02-01 and DGA E-12/25, and by FEDER funds.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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