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Euler Polynomial

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EulerE

The Euler polynomial E_n(x) is given by the Appell sequence with

g(t)=1/2(e^t+1),
(1)

giving the generating function

(2e^(xt))/(e^t+1)=sum_(n=0)^inftyE_n(x)(t^n)/(n!).
(2)

The first few Euler polynomials are

E_0(x)=1
(3)
E_1(x)=x-1/2
(4)
E_2(x)=x^2-x
(5)
E_3(x)=x^3-3/2x^2+1/4
(6)
E_4(x)=x^4-2x^3+x
(7)
E_5(x)=x^5-5/2x^4+5/2x^2-1/2.
(8)

Roman (1984, p. 100) defines a generalization E_n^((alpha))(x) for which E_n(x)=E_n^((1))(x). Euler polynomials are related to the Bernoulli numbers by

E_(n-1)(x)=(2^n)/n[B_n((x+1)/2)-B_n(x/2)]
(9)
=2/n[B_n(x)-2^nB_n(x/2)]
(10)
E_(n-2)(x)=2(n; 2)^(-1)sum_(k=0)^(n-2)(n; k)[(2^(n-k)-1)B_(n-k)B_k(x)],
(11)

where (n; k) is a binomial coefficient. Setting x=1/2 and normalizing by 2^n gives the Euler number

E_n=2^nE_n(1/2).
(12)

The first few values of E_n(0) are -1/2, 0, 1/4, -1/2, 0, 17/8, 0, 31/2, 0, .... The terms are the same but with the signs reversed if x=1. 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)].
(13)

The Bernoulli numbers B_n for n>1 can be expressed in terms of E_n(0) by

B_n=-(nE_(n-1)(0))/(2(2^n-1)).
(14)

The Newton expansion of the Euler polynomials is given by

E_n(x)=sum_(j=0)^nsum_(k=j)^n(-1; j)1/(2^j)(k)_jS(n,k)(x)_(k-j),
(15)

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

The Euler polynomials satisfy the identities

E_n(x+1)+E_n(x)=2x^n
(16)

and

sum_(k=0)^n(n; k)E_k(z)E_(n-k)(w)=2(1-w-z)E_n(z+w)+2E_(n+1)(z+w)
(17)

for n a nonnegative integer.

See also

Appell Sequence, Bernoulli Polynomial, Euler Number, Genocchi Number, Prime-Generating Polynomial

Related Wolfram sites

https://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 zeta(s,x), Bernoulli Polynomials B_n(x), Euler Polynomials E_n(x), and Polylogarithms Li_nu(x)." §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 E_n(x)." Ch. 20 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 175-181, 1987.

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Euler Polynomial

Cite this as:

Weisstein, Eric W. "Euler Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulerPolynomial.html

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