Euler polynomials

Polynomials of the form

where are the Euler numbers. The Euler polynomials can be computed successively by means of the formula

In particular,

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.

References

[1]	L. Euler, "Opera omnia: series prima: opera mathematica: institutiones calculi differentialis" , Teubner (1980) (Translated from Latin)
[2]	N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924)

Comments

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 ,