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

Adell, J.; Jodrá, P.

doi:10.1090/S0002-9947-07-04411-X

On a Ramanujan equation connected with the median of the gamma distribution
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by J. A. Adell and P. Jodrá PDF

Trans. Amer. Math. Soc. 360 (2008), 3631-3644 Request permission

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

José A. Adell and Pedro Jodrá, Sharp estimates for the median of the $\Gamma (n+1,1)$ distribution, Statist. Probab. Lett. 71 (2005), no. 2, 185–191. MR 2126774, DOI 10.1016/j.spl.2004.10.025
J. A. Adell and P. Jodrá, On the complete monotonicity of a Ramanujan sequence connected with $e^n$, to appear in The Ramanujan Journal.
José A. Adell and Alberto Lekuona, Taylor’s formula and preservation of generalized convexity for positive linear operators, J. Appl. Probab. 37 (2000), no. 3, 765–777. MR 1782452, DOI 10.1017/s0021900200015989
José Antonio Adell and Alberto Lekuona, Sharp estimates in signed Poisson approximation of Poisson mixtures, Bernoulli 11 (2005), no. 1, 47–65. MR 2121455, DOI 10.3150/bj/1110228242
Sven Erick Alm, Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence, Bernoulli 9 (2003), no. 2, 351–371. MR 1997033, DOI 10.3150/bj/1068128981
Horst Alzer, On Ramanujan’s inequalities for $\exp (k)$, J. London Math. Soc. (2) 69 (2004), no. 3, 639–656. MR 2050038, DOI 10.1112/S0024610704005253
Horst Alzer, Proof of the Chen-Rubin conjecture, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 4, 677–688. MR 2173334, DOI 10.1017/S0308210500004066
C. Berg and H. L. Pedersen, The Chen-Rubin conjecture in a continuous setting. Preprint.
Bruce C. Berndt, Ramanujan’s notebooks. Part II, Springer-Verlag, New York, 1989. MR 970033, DOI 10.1007/978-1-4612-4530-8
Bruce C. Berndt, Youn-Seo Choi, and Soon-Yi Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, Continued fractions: from analytic number theory to constructive approximation (Columbia, MO, 1998) Contemp. Math., vol. 236, Amer. Math. Soc., Providence, RI, 1999, pp. 15–56. MR 1665361, DOI 10.1090/conm/236/03488
Paul Bracken, A function related to the central limit theorem, Comment. Math. Univ. Carolin. 39 (1998), no. 4, 765–775. MR 1715465
Jeesen Chen and Herman Rubin, Bounds for the difference between median and mean of gamma and Poisson distributions, Statist. Probab. Lett. 4 (1986), no. 6, 281–283. MR 858317, DOI 10.1016/0167-7152(86)90044-1
Tseng Tung Cheng, The normal approximation to the Poisson distribution and a proof of a conjecture of Ramanujan, Bull. Amer. Math. Soc. 55 (1949), 396–401. MR 29487, DOI 10.1090/S0002-9904-1949-09223-6
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
K. P. Choi, On the medians of gamma distributions and an equation of Ramanujan, Proc. Amer. Math. Soc. 121 (1994), no. 1, 245–251. MR 1195477, DOI 10.1090/S0002-9939-1994-1195477-8
E. T. Copson, An approximation connected with $e^{-x}$, Proc. Edinburgh Math. Soc. (2) 3 (1933), 201–206.
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Philippe Flajolet, Peter J. Grabner, Peter Kirschenhofer, and Helmut Prodinger, On Ramanujan’s $Q$-function, J. Comput. Appl. Math. 58 (1995), no. 1, 103–116. MR 1344359, DOI 10.1016/0377-0427(93)E0258-N
Norman L. Johnson, Samuel Kotz, and Adrienne W. Kemp, Univariate discrete distributions, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR 1224449
Jonathan M. Kane, Monotonic approach to central limits, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2127–2133. MR 1825926, DOI 10.1090/S0002-9939-00-05776-2
J. Karamata, Sur quelques problèmes posés par Ramanujan, J. Indian Math. Soc. (N.S.) 24 (1960), 343–365 (1961) (French). MR 133307
Eric S. Key, A probabilistic approach to a conjecture of Ramanujan, J. Ramanujan Math. Soc. 4 (1989), no. 2, 109–119. MR 1040203
J. C. W. Marsaglia, The incomplete Gamma function and Ramanujan’s rational approximation to $e^x$, J. Statis. Comput. Simul. 24 (1986), 163–169.
R. B. Paris, On a generalisation of a result of Ramanujan connected with the exponential series, Proc. Edinburgh Math. Soc. (2) 24 (1981), no. 3, 179–195. MR 633719, DOI 10.1017/S0013091500016503
S. Ramanujan, Question 294, J. Indian Math. Soc. 3 (1911), 128.
S. Ramanujan, On question 294, J. Indian Math. Soc. 4 (1912), 151–152.
S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
Bero Roos, Improvements in the Poisson approximation of mixed Poisson distributions, J. Statist. Plann. Inference 113 (2003), no. 2, 467–483. MR 1965122, DOI 10.1016/S0378-3758(02)00095-2
G. Szegö, Über einige von S. Ramanujan gestelle Aufgaben, J. London Math. Soc. 3 (1928), 225–232.
Henry Teicher, An inequality on Poisson probabilities, Ann. Math. Statist. 26 (1955), 147–149. MR 67384, DOI 10.1214/aoms/1177728608
G. N. Watson, Theorems stated by Ramanujan (V): approximations connected with $e^x$, Proc. London Math. Soc. 29 (1929), 293–308.