On the Hurwitz Zeta Function and Its Applications to Hyperbolic Probability Distributions
Tsung-Lin Cheng, Chin-Yuan Hu
TL;DR
This paper presents new proofs and formulas related to the Hurwitz zeta function, including applications to hyperbolic probability distributions, and explores the properties of sinh and tanh distributions.
Contribution
It introduces a novel proof of Jensen's formula, derives new formulas for the generalized Bernoulli function, and investigates hyperbolic probability density functions.
Findings
New proof of Jensen's formula from 1895
Formulas similar to Pitman and Yor (2003) derived
Probability density functions of sinh and tanh distributions analyzed
Abstract
In this paper, we propose a new proof of the Jensen formula in 1895. We also derive some formulas similar to those in Pitman and Yor, 2003. Besides, a new formula of the generalized Bernoulli function is also derived. At the end of the paper, the probability density functions of sinh and tanh are studied briefly for general cases.
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Taxonomy
TopicsStatistical Distribution Estimation and Applications · Mathematical functions and polynomials · Advanced Statistical Methods and Models