Differential calculus for linear operators represented by finite signed measures and applications

Lekuona

2017

J Inequal Appl

We construct sequences of finite sums and converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant γ 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 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.

“…Finally, let be a differentiable function such that , , where E stands for mathematical expectation. It has been shown in [ 18 ], Theorem 7.1, that …”

Section: Auxiliary Resultsmentioning

confidence: 99%

Monotone and fast computation of Euler’s constant

Lekuona

2017

J Inequal Appl

“…Proof. Taking into account Lemmas 8.1 and 8.3 above, the proof follows along the lines of that in Theorem 8.2 in [1].…”

mentioning

confidence: 85%

Erratum to: Differential calculus for linear operators represented by finite signed measures and applications

2012

Acta Math Hung

Self Cite

“…Finally, we mention that formula (1.3) is one example of the differential calculus for linear operators represented by stochastic processes developed in [4]. As we will see in the following sections, this probabilistic methodology gives a unified approach to deal with various problems in analytic number theory.…”

Section: Introductionmentioning

confidence: 96%

“…The aim of this paper is to give 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 the differential calculus for the negative binomial process developed in [4].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Acceleration Methods for Series: A Probabilistic Perspective

Lekuona

2016

Mediterr. J. Math.

Self Cite

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.