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

Authors:	J A Adell P Jodrá

Institution:	Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009 Zaragoza, Spain ; Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009 Zaragoza, Spain

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 $(\lambda_n)_{n\ge 1}$ .

Keywords:	Central limit theorem Charlier polynomials forward difference gamma distribution median Poisson process Ramanujan's equation

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