On a Ramanujan equation connected with the median of the gamma distribution
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- by J. A. Adell and P. Jodrá PDF
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Abstract:
In this paper, we consider the sequence $(\theta _n)_{n\ge 0}$ solving the Ramanujan equation \[ \frac {e^n}{2}=\sum _{k=0}^{n}\frac {n^k}{k!}+\frac {n^n}{n!} (\theta _n-1),\qquad n=0,1,\dots . \] 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 $(\lambda _n)_{n\ge 1}$.References
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Additional Information
- J. A. Adell
- Affiliation: Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009 Zaragoza, Spain
- MR Author ID: 340766
- Email: adell@unizar.es
- P. Jodrá
- Affiliation: Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009 Zaragoza, Spain
- Email: pjodra@unizar.es
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2386240