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

Abstract

Abstract. In this paper, we consider the sequence (θ n ) n≥0 solving the Ramanujan equationThe three main achievements are the following. We introduce a continuoustime extension θ(t) of θ n and show its close connections with the medians λ n of the Γ(n + 1, 1) distributions and the Charlier polynomials. We give upper and lower bounds for both θ(t) and λ n , in particular for θ 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 (λ n ) n≥1 .

Computing the Asymptotic Expansion of the Median of the Erlang Distribution

“…Now, we answer question (iii) related to the bounds of the median of the Erlang distribution. As it was said in Section 1, Adell and Jodrá [2] have given upper and lower bounds for λ n , specifically λ 6 (n) < λ n < λ 7 (n) for n ≥ 1. Moreover, Adell and Jodrá [2,Section 4] remark that the same methodology can be used to obtain sharper bounds for λ n if we consider more terms in the asymptotic expansion of λ n .…”

“…. More specifically, Adell and Jodrá [2] noticed that c k (n) = C k (n; n) so that the coefficients c k (n) in Eq. (2.1) can also be computed by means of Eq.…”

“…With the aim of refining the bounds given in Eq. (1.1), Adell and Jodrá [2] computed the first seven coefficients q j in the asymptotic expansion of λ n (see Table 1) and then they derived the following upper and lower bounds for λ n :…”

“…(ii) Which is the partial sum of the asymptotic expansion of λ n closest to the true value of λ n as well as its number of significant digits? (iii) Can sharper bounds for λ n be obtained by considering more terms in the asymptotic expansion of λ n as it is proposed in Adell and Jodrá [2]?…”

Asymptotic estimates for Stieltjes constants: a probabilistic approach

“…We mention that such a differential calculus has already found applications in dealing with estimates of the remainder of a certain Ramanujan series connected with the median of the gamma distribution (cf. Adell & Jodrá 2008), as well as estimates of the entropy of the Poisson law in an information theory setting (cf. Adell et al 2010), among other applications.…”