19.2 Euler Numbers and Euler Polynomials
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19.2 Euler Numbers and Euler Polynomials
19.2.1 Equations
Some Euler numbers and Euler polynomials are, e.g., listed in [1, 2] as well as most
of the following equations.
The generating function for the Euler polynomials E n (x) reads
2 exp(x t)
exp(t) + 1
=
∞
n=0
E n (x)
t n
n!
.
(19.7)
Thus, e.g.,
E 0 (x) = 1, E 1 (x) = x −
1
2
, E 2 (x) = x
2
− x, · · · .
(19.8)
The Euler numbers E n can be derived from the corresponding polynomials via
E n = 2
n E n
1
2
, e.g., E 0 = 1, E 2 = −1, E 4 = 5 and E 2n+1 = 0 for n ≥ 0.
(19.9)
The derivative of the Euler polynomial follows the same relation as the formula for
the Bernoulli polynomials:
d
dx
E n (x) = n E n−1 (x),
(19.10)
and therefore if we know the integration constant it will be straightforward to
compute Euler polynomials by direct integration. For polynomials of odd degree
this constant will be 0. The polynomial values between 0 ≤ x ≤ 1 can be evaluated
by
E 2n (x) =
4(−1) n (2n)!
π 2n+1
∞
k=0
sin((2k + 1)πx)
(2k + 1) 2n+1 , n > 0,
(19.11a)
E 2n−1 (x) =
4(−1) n (2n − 1)!
π 2n+1
∞
k=0
cos((2k + 1)πx)
(2k + 1) 2n , n ≥ 1. (19.11b)
For sufficiently large n values these series will quickly converge due to the
denominator.
E n (−x) = (−1)
n+1
[E n (x) − 2x
n
]
(19.12a)