Euler Numbers

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Euler Numbers


in mathematics, the integers En that are the coefficients of tn/n! in the expansion of the function 1/cosh t in the power series

These numbers, which were introduced by L. Euler in 1755, are related by the formula (E + 1)n + (E – 1)n = 0, n = 1, 2, 3, . . ., E0 = 1 (after raising to the power, Ek must be inserted for Ek). The following relations hold between Euler numbers and Bernoulli numbers:

Euler numbers are encountered in various formulas of mathematical analysis.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
When x = 0, [E.sub.n] = [E.sub.n](0)(n [greater than or equal to] 0) are called the Euler numbers.
It is well-known that numbers of up-down permutations are Euler numbers. Suppose E(n) is the nth Euler number, then |[C[C.sub.=2[n-1]] | = E(2n).
Kim, "Euler numbers and polynomials associated with zeta functions," Abstract and Applied Analysis, vol.
Bernoulli numbers and Euler numbers are [B.sub.k] = [B.sub.k](0) and [E.sub.k] = [2.sup.k][E.sub.k](1/2).
We follow the approach of Ohser and Schladitz (2009) based on a Crofton formula which boils down the estimation of the integral of the mean curvature to computing Euler numbers in virtual planar sections.
After introducing the Bernoulli numbers, Euler introduced the Euler numbers to study the sum [T.sub.k] (n) = [[summation of].sup.n-1.sub.r=0] [(-1).sup.r] [r.sup.k].
The inverse of 1 + [U.sup.h] can be explicitly expressed using Euler numbers and polynomials, [4, Eq.
Another approach to quotient sets became rather popular after Vafa and Witten introduced in [17] the so called stringy Euler numbers. In a nutshell they considered the Euler numbers of orbiquotient sets
The Euler numbers and flower numbers for Plot A also showed similar temporal trends (Fig.