Sharp Bounds on the Entropy of the Poisson Law and Related Quantities

One of the difficulties in calculating the capacity of certain Poisson channels is that H(lambda), the entropy of the Poisson distribution with mean lambda, is not available in a simple form. In this work we derive upper and lower bounds for H(lambda) that are asymptotically tight and easy to compute. The derivation of such bounds involves only simple probabilistic and analytic tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on the relative entropy D(n, p) between a binomial and a Poisson, thus refining the work of Harremoes and Ruzankin (2004). Bounds on the entropy of the binomial also follow easily.Comment: To appear, IEEE Trans. Inform. Theor

Series acceleration via negative binomial probabilities

Many special functions and analytic constants allow for a probabilistic representation in terms of inverse moments of [0, 1]-valued random variables. Under this assumption, we give fast computations of them with an explicit upper bound for the remainder term. One of the main features of the method is that the coefficients of the main term of the approximation always contain negative binomial probabilities which, in turn, can be precomputed and stored. Applications to the arctangent function, Dirichlet functions and their nth derivatives, and the Catalan, Gompertz, and Stieltjes constants are provided

Monotone and fast computation of Euler’s constant

We construct sequences of finite sums (l˜n)n=0 and (u˜n)n=0 converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant ¿ at the geometric rate 1/2. Such sequences are easy to compute and satisfy complete monotonicity-type properties. As a consequence, we obtain an infinite product representation for 2 ¿ converging in a monotone and fast way at the same time. We use a probabilistic approach based on a differentiation formula for the gamma process

Binomial Identities and Moments of Random Variables

We give unified simple proofs of some binomial identities, by using an elementary identity on moments of random variables

Explicit Expressions for Higher Order Binomial Convolutions of Numerical Sequences

We give explicit expressions for higher order binomial convolutions of sequences of numbers having a finite exponential generating function. Illustrations involving Cauchy, Bernoulli, and Apostol–Euler numbers are presented. In these cases, we obtain formulas easy to compute in terms of Stirling numbers

Acceleration Methods for Series: A Probabilistic Perspective

We introduce a probabilistic perspective to the problem of accelerating the convergence of a wide class of series, paying special attention to the computation of the coefficients, preferably in a recursive way. This approach is mainly based on a differentiation formula for the negative binomial process which extends the classical Euler’s transformation. We illustrate the method by providing fast computations of the logarithm and the alternating zeta functions, as well as various real constants expressed as sums of series, such as Catalan, Stieltjes, and Euler–Mascheroni constants

A unified approach to higher order convolutions within a certain subset of appell polynomials

We consider the subset R of Appell polynomials whose exponential generating function is given in terms of the moment generating function of a certain random variable Y. This subset contains the Hermite, Bernoulli, Apostol–Euler, and Cauchy type polynomials, as well as various kinds of their generalizations, among others. We obtain closed form expressions for higher order convolutions of Appell polynomials in the subset R. We give a unified approach mainly based on random scale transformations of Appell polynomials, as well as on a probabilistic generalization of the Stirling numbers of the second kind. Different illustrative examples, including reformulations of convolution identities already known in the literature, are discussed in detail. In such examples, the convolution identities involve the classical Stirling numbers