Polynomials of the form
where are the Euler numbers. The Euler polynomials can be computed successively by means of the formula
The Euler polynomials satisfy the difference equation
and belong to the class of Appell polynomials, that is, they satisfy
The generating function of the Euler polynomials is
The Euler polynomials admit the Fourier expansion
They satisfy the relations
if is odd,
if is even. Here is a Bernoulli polynomial (cf. Bernoulli polynomials). The periodic functions coinciding with the right-hand side of (*) are extremal in the Kolmogorov inequality and in a number of other extremal problems in function theory. Generalized Euler polynomials have also been considered.
|||L. Euler, "Opera omnia: series prima: opera mathematica: institutiones calculi differentialis" , Teubner (1980) (Translated from Latin)|
|||N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924)|
The Euler polynomials satisfy in addition the identities
written symbolically as
Here the right-hand side should be read as follows: first expand the right-hand side into sums of expressions and then replace with .
Using the same symbolic notation one has for every polynomial ,
Euler polynomials. Yu.N. Subbotin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Euler_polynomials&oldid=17907