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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): J. A. Adell; P. Jodrá
Journal: Trans. Amer. Math. Soc. 360 (2008), 3631-3644.
MSC (2000): Primary 41A60; Secondary 60E05
Posted: 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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