Euler Numbers

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.

References in periodicals archive ?
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.
In fact, the computation of the Euler number in virtual planar sections can be replaced by counting connected components or tangent points, as suggested in Schladitz etal.
After introducing the Bernoulli numbers, Euler introduced the Euler numbers to study the sum [T.
These classical integral geometric formulae allow to compute the intrinsic volumes by calculation of Euler numbers in lower dimensional intersections and subsequent integration over all positions of the intersecting affine subspaces.
Lehmer in 1934 extended these methods to Euler numbers, Genocchi numbers, and Lucas numbers (1934) [9], and calculated the 196-th Bernoulli numbers.
Kurt, p-adic interpolation functions and Kummer-type congruences for q-twisted and q-generalized twisted Euler numbers, Adv.
The Crofton formulae reduce measurement of the intrinsic volumes to computation of Euler numbers in lower dimensional sections.
Abstract In this paper discussed recurrence formula and sum formulas for the generalised Euler numbers [E.
In particular, the Euler number of the background differs from the sum of the Euler numbers of the obtained equivalence classes.
Abstract In this paper, we prove some new recurrence formulas for the generalized Euler numbers [E.
Abstract In this paper, we establish some recurrence formulas for generalized Euler numbers.
Abstract The main purpose of this paper is using the elementary method to obtain some interesting identities involving the Bernoulli numbers and the Euler numbers.