On a Ramanujan Equation Connected with the Median of the Gamma Distribution
نویسنده
چکیده
In this paper, we consider the sequence (θn)n≥0 solving the Ramanujan equation en 2 = n ∑ k=0 nk k! + nn n! (θn − 1), n = 0, 1, . . . . The three main achievements are the following. We introduce a continuous– time extension θ(t) of θn and show its close connections with the medians λn of the Γ(n+1, 1) distributions and the Charlier polynomials. We give upper and lower bounds for both θ(t) and λn, in particular for θn, 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 (λn)n≥1.
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تاریخ انتشار 2008