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.