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Probabilistic Proofs of Euler Identities

Part of: Elementary classical functions History of mathematics and mathematicians

Published online by Cambridge University Press:  30 January 2018

Lars Holst
Lars Holst*
Affiliation:
Royal Institute of Technology
*
∗ Postal address: Department of Mathematics, Royal Institute of Technology, SE-10044 Stockholm, Sweden. Email address: lholst@math.kth.se
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Abstract

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Formulae for ζ(2n) and Lχ4(2n + 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2 / 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

Keywords

Basel problemhyperbolic secant distributionEuler numbertangent numberEuler's sine product

MSC classification

Primary: 33B10: Exponential and trigonometric functions
Secondary: 01A50: 18th century
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1206 - 1212
Copyright
© Applied Probability Trust 

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