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Euler numbers


Euler numbersDlmfDlmfPlanetmathPlanetmath En have the generating function 1coshx such that

1coshx=:n=0Enn!xn.

They are integers but have no expression for calculating them.  Their only are that the numbers with odd index (http://planetmath.org/IndexingSet) are all 0 and that

sgn(E2m)=(-1)m  

The Euler number have intimate relation to the Bernoulli numbersDlmfDlmfMathworldPlanetmathPlanetmath.  The first Euler numbers with even index are

E0=1,E2=-1,E4=5,E6=-61,E8=1385,E10=-50521.
  • One can by hand determine Euler numbers by performing the division of 1 by the Taylor seriesMathworldPlanetmath of hyperbolic cosineMathworldPlanetmath (cf. Taylor series via division and Taylor series of hyperbolic functions).  Since  coshix=cosx,  the division 1:cosx correspondingly gives only terms with plus sign, i.e. it shows the absolute valuesMathworldPlanetmathPlanetmathPlanetmath of the Euler numbers.

  • The Euler numbers may also be obtained by using the Euler polynomialsMathworldPlanetmath En(x):

    En= 2nEn(12)
  • If the Euler numbers Ek are denoted as symbolic powers Ek, then one may write the equation

    (E+1)n+(E-1)n= 0,

    which can be used as a recurrence relation for computing the values of the even-indexed Euler numbers.  Cf. the Leibniz rulePlanetmathPlanetmath for derivatives of product fg.

Title Euler numbers
Canonical name EulerNumbers
Date of creation 2014-12-02 17:43:40
Last modified on 2014-12-02 17:43:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Definition
Classification msc 11B68
Related topic GudermannianFunction
Related topic BernoulliNumber
Related topic InverseGudermannianFunction
Related topic HermiteNumbers