Euler Polynomial

The Euler polynomial  is given by the Appell sequence with

(1)

giving the generating function

(2)

The first few Euler polynomials are

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

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

			(9)
			(10)
			(11)

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.

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.