The Euler polynomial
is given by the Appell sequence with

(1)

giving the generating function

(2)

The first few Euler polynomials are

Roman (1984, p. 100) defines a generalization for which . Euler polynomials are related to the Bernoulli numbers by

where
is a binomial coefficient . Setting and normalizing by gives the Euler number

(12)

The first few values of are , 0, 1/4, , 0, 17/8, 0, 31/2, 0, .... The terms are the same but with
the signs reversed if . These values can be computed using the double
series

(13)

The Bernoulli numbers for can be expressed in terms of by

(14)

The Newton expansion of the Euler polynomials is given by

(15)

where
is a binomial coefficient , is a falling factorial ,
and
is a Stirling number of the second
kind (Roman 1984, p. 101).

The Euler polynomials satisfy the identities

(16)

and

(17)

for
a nonnegative integer .

See also Appell Sequence ,

Bernoulli Polynomial ,

Euler Number ,

Genocchi
Number ,

Prime-Generating Polynomial
Related Wolfram sites http://functions.wolfram.com/Polynomials/EulerE2/
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References Abramowitz, M. and Stegun, I. A. (Eds.). "Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula." §23.1 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 804-806, 1972. Gradshteyn, I. S. and Ryzhik,
I. M. Tables
of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
2000. Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A.
"The Generalized Zeta Function , Bernoulli Polynomials , Euler Polynomials , and Polylogarithms ." §1.2 in Integrals
and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach,
pp. 23-24, 1990. Roman, S. "The Euler Polynomials." §4.2.3
in The
Umbral Calculus. New York: Academic Press, pp. 100-106, 1984. Spanier,
J. and Oldham, K. B. "The Euler Polynomials ." Ch. 20 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 175-181, 1987. Referenced
on Wolfram|Alpha Euler Polynomial
Cite this as:
Weisstein, Eric W. "Euler Polynomial."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/EulerPolynomial.html

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