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Euler Number

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The Euler numbers, also called the secant numbers or zig numbers, are defined for |x|<pi/2 by

sechx-1=-(E_1^*x^2)/(2!)+(E_2^*x^4)/(4!)-(E_3^*x^6)/(6!)+...
(1)
secx-1=(E_1^*x^2)/(2!)+(E_2^*x^4)/(4!)+(E_3^*x^6)/(6!)+...,
(2)

where sech(z) is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.

A different sort of Euler number, the Euler number of a finite complex K, is defined by

chi(K)=sum(-1)^prank(C_p(K)).
(3)

This Euler number is a topological invariant.

To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial n^2-n+41 are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes, and prime Euler numbers are terms Euler number primes.

Some values of the (secant) Euler numbers are

E_1^*=1
(4)
E_2^*=5
(5)
E_3^*=61
(6)
E_4^*=1385
(7)
E_5^*=50521
(8)
E_6^*=2702765
(9)
E_7^*=199360981
(10)
E_8^*=19391512145
(11)
E_9^*=2404879675441
(12)
E_(10)^*=370371188237525
(13)
E_(11)^*=69348874393137901
(14)
E_(12)^*=15514534163557086905
(15)

(OEIS A000364).

The slightly different convention defined by

E_(2n)=(-1)^nE_n^*
(16)
E_(2n+1)=0
(17)

is frequently used. These are, for example, the Euler numbers computed by the Wolfram Language function EulerE[n]. This definition has the particularly simple series definition

sechx=sum_(k=0)^infty(E_kx^k)/(k!)
(18)

and is equivalent to

E_n=2^nE_n(1/2),
(19)

where E_n(x) is an Euler polynomial.

The number of decimal digits in E_n for n=0, 2, 4, ... are 1, 1, 1, 2, 4, 5, 7, 9, 11, 13, 15, 17, ... (OEIS A047893). The number of decimal digits in E_(10^n) for n=0, 1, ... are 1, 5, 139, 2372, 33699, ... (OEIS A103235).

The Euler numbers have the asymptotic series

E_(2n)∼(-1)^n8sqrt(n/pi)((4n)/(pie))^(2n).
(20)

A more efficient asymptotic series is given by

E_(2n)∼(-1)^n8sqrt(n/pi)((4n)/(pie)(480n^2+9)/(480n^2-1))^(2n)
(21)

(P. Luschny, pers. comm., 2007).

Expanding (E-i)^n for even n gives the identity

(E-i)^n={0 for n even; -iT_((n+1)/2) for n odd.
(22)

where the coefficient E^n is interpreted as |E_n| (Ely 1882; Fort 1948; Trott 2004, p. 69) and T_n is a tangent number.

Stern (1875) showed that

E_k=E_l (mod 2^n)
(23)

iff k=l (mod 2^n). This result had been previously stated by Sylvester in 1861, but without proof.

Shanks (1968) defines a generalization of the Euler numbers by

c_(a,n)=((2n)!L_a(2n+1))/(sqrt(a))((2a)/pi)^(2n+1).
(24)

Here,

c_(1,n)=1/2(-1)^nE_(2n),
(25)

and c_(2,n) is (2n)! times the coefficient of x^(2n) in the series expansion of cosx/cos(2x). A similar expression holds for c_(3,n), but strangely not for c_(a,n) with a>=4. The following table gives the first few values of c_(a,n) for n=0, 1, ....

aOEISc_(a,n)
1A0003641, 1, 5, 61, ...
2A0002811, 3, 57, 2763, ...
3A0004361, 8, 352, 38528, ...
4A0004901, 16, 1280, 249856, ...
5A0001872, 30, 3522, 1066590, ...
6A0001922, 46, 7970, 3487246, ...
7A0640681, 64, 15872, 9493504, ...
8A0640692, 96, 29184, 22634496, ...
9A0640702, 126, 49410, 48649086, ...
10A0640712, 158, 79042, 96448478, ...

See also

Bernoulli Number, Euler Characteristic, Euler Number Prime, Euler Prime, Eulerian Number, Euler Polynomial, Euler Zigzag Number, Genocchi Number, Integer Sequence Primes, Lefschetz Number, Prime-Generating Polynomial, Tangent Number

Related Wolfram sites

http://functions.wolfram.com/IntegerFunctions/EulerE/

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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.Caldwell, C. "The Top 20: Euler Irregular." http://primes.utm.edu/top20/page.php?id=25.Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 110-111, 1996.Ely, G. S. "Some Notes on the Numbers of Bernoulli and Euler." Amer. J. Math. 5, 337-341, 1882.Fort, T. Finite Differences and Difference Equations in the Real Domain. Oxford, England: Clarendon Press, 1948.Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, p. 47, 2000.Guy, R. K. "Euler Numbers." §B45 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 101, 1994.Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen für Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995.Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663-688, 1967.Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub.,p. 124, 1993.Shanks, D. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967.Shanks, D. Corrigendum to "Generalized Euler and Class Numbers." Math. Comput. 22, 699, 1968.Sloane, N. J. A. Sequences A000364/M4019, A014547, A047893, A092823, A103234, and A103235 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Euler Numbers, E_n." Ch. 5 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 39-42, 1987.Stern, M. A. "Zur Theorie der Euler Schen Zahlen." J. reine angew. Math. 79, 67-98, 1875.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.Young, P. T. "Congruences for Bernoulli, Euler, and Stirling Numbers." J. Number Th. 78, 204-227, 1999.

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Euler Number

Cite this as:

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

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