Euler numbers

The coefficients $E_n$ in the expansion

$$\frac1{\cosh z}=\sum_{n=0}^\infty E_n\frac{z^n}{n!}.$$

The recurrence formula for the Euler numbers ($E^n\equiv E_n$ in symbolic notation) has the form

$$(E+1)^n+(E-1)^n=0,\quad E_0=1.$$

Thus, $E_{2n+1}=0$, the $E_{4n}$ are positive and the $E_{4n+2}$ are negative integers for all $n=0,1,\dots$; $E_2=-1$, $E_4=5$, $E_6=-61$, $E_8=1385$, and $E_{10}=-50521$. The Euler numbers are connected with the Bernoulli numbers $B_n$ by the formulas

$$E_{n-1}=\frac{(4B-1)^n-(4B-3)^n}{2n},$$

$$E_{2n}=\frac{4^{2n+1}}{2n+1}\left(B-\frac14\right)^{2n+1}.$$

The Euler numbers are used in the summation of series. For example,

$$\sum_{k=0}^\infty(-1)^k\frac1{(2k+1)^{2n+1}}=\frac{\pi^{2n+1}}{2^{2n+2}(2n)!}|E_{2n}|.$$

Sometimes the $|E_{2n}|$ are called the Euler numbers.

These numbers were introduced by L. Euler (1755).

References

[1]	L. Euler, "Institutiones calculi differentialis" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 10 , Teubner (1980)
[2]	I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products" , Acad. Press (1980) (Translated from Russian)

Comments

The symbolic formula $(E+1)^n+(E-1)^n=0$ should be interpreted as follows: first expand the left-hand side as a sum of the powers $E^m$, then replace $E^m$ with $E_m$. Similarly for the formula connecting the Bernoulli and Euler numbers. The Euler numbers $E_n$ are obtained from the Euler polynomials $E_n(x)$ by $E_n=2^nE_n(1/2)$.

References

[a1]	A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)