Euler Polynomial
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"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
الجزء والصفحة:
...
17-9-2019
2917
Euler Polynomial
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
![E_n(0)=2^(-n)sum_(j=1)^n[(-1)^(j+n+1)j^nsum_(k=0)^(n-j)(n+1; k)].](http://mathworld.wolfram.com/images/equations/EulerPolynomial/NumberedEquation4.gif) |
(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.
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