Probabilistic Proofs of Euler Identities

DOI10.1239/JAP/1389370108zbMath1295.33002OpenAlexW2008284771MaRDI QIDQ5407038

Lars Holst

Publication date: 4 April 2014

Published in: Journal of Applied Probability (Search for Journal in Brave)

Full work available at URL: https://projecteuclid.org/euclid.jap/1389370108

zbMATH Keywords

Euler numberBasel problemtangent numberhyperbolic secant distributionEuler's sine product

Mathematics Subject Classification ID

Central limit and other weak theorems (60F05) Probability distributions: general theory (60E05) History of mathematics in the 18th century (01A50) Exponential and trigonometric functions (33B10)

Related Items (1)

Three systems of orthogonal polynomials and \(L^{2}\)-boundedness of two associated operators

Cites Work

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• Euler's formulae for \(\zeta(2n)\) and products of Cauchy variables
• Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function
• A Short Proof of ζ(2) = π²/6
• Probabilistically Proving that ζ(2) = π²⁄6
• A Simple Proof of 1 + 1 2 2 + 1 3 2 + ⋯ = π 2 6 and Related Identities
• Another Simple Proof of 1 + 1 2 2 + 1 3 2 + ⋅⋅⋅ = π 2 6
• The probability law for the sum of 𝑛 independent variables, each subject to the law (1/\vphantom{1(2ℎ)}.\kern-\nulldelimiterspace(2ℎ))𝑠𝑒𝑐ℎ(𝜋𝑥/\vphantom{𝜋𝑥(2ℎ)}.\kern-\nulldelimiterspace(2ℎ))
• A Stochastic Approach to the Gamma Function
• Proof of Euler’s Infinite Product for the Sine
• Generalized Hyperbolic Secant Distributions
• Six Ways to Sum a Series

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