866 research outputs found
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
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
Explicit expressions and integral representations for the Stirling numbers. A probabilistic approach
We show that the Stirling numbers of the first and second kind can be represented in terms of moments of appropriate random variables. This allows us to obtain, in a unified way, a variety of old and new explicit expressions and integral representations of such numbers
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