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The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365401.png" /> in the expansion
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{{TEX|done}}
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The coefficients $E_n$ in the expansion
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365402.png" /></td> </tr></table>
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$$\frac1{\cosh z}=\sum_{n=0}^\infty E_n\frac{z^n}{n!}.$$
  
The recurrence formula for the Euler numbers (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365403.png" /> in symbolic notation) has the form
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The recurrence formula for the Euler numbers ($E^n\equiv E_n$ in symbolic notation) has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365404.png" /></td> </tr></table>
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$$(E+1)^n+(E-1)^n=0,\quad E_0=1.$$
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365405.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365406.png" /> are positive and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365407.png" /> are negative integers for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365408.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e0365409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654013.png" />. The Euler numbers are connected with the [[Bernoulli numbers|Bernoulli numbers]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654014.png" /> by the formulas
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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|Bernoulli numbers]] $B_n$ by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654015.png" /></td> </tr></table>
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$$E_{n-1}=\frac{(4B-1)^n-(4B-3)^n}{2n},$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654016.png" /></td> </tr></table>
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$$E_{2n}=\frac{4^{2n+1}}{2n+1}\left(B-\frac14\right)^{2n+1}.$$
  
 
The Euler numbers are used in the summation of series. For example,
 
The Euler numbers are used in the summation of series. For example,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654017.png" /></td> </tr></table>
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$$\sum_{k=0}^\infty(-1)^k\frac1{(2k+1)^{2n+1}}=\frac{\pi^{2n+1}}{2^{2n+2}(2n)!}|E_{2n}|.$$
  
Sometimes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654018.png" /> are called the Euler numbers.
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Sometimes the $|E_{2n}|$ are called the Euler numbers.
  
 
These numbers were introduced by L. Euler (1755).
 
These numbers were introduced by L. Euler (1755).
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====Comments====
 
====Comments====
The symbolic formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654019.png" /> should be interpreted as follows: first expand the left-hand side as a sum of the powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654020.png" />, then replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654021.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654022.png" />. Similarly for the formula connecting the Bernoulli and Euler numbers. The Euler numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654023.png" /> are obtained from the [[Euler polynomials|Euler polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654024.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654025.png" />.
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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|Euler polynomials]] $E_n(x)$ by $E_n=2^nE_n(1/2)$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Segun,  M. Abramowitz,  "Handbook of mathematical functions" , ''Appl. Math. Ser.'' , '''55''' , Nat. Bur. Standards  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Segun,  M. Abramowitz,  "Handbook of mathematical functions" , ''Appl. Math. Ser.'' , '''55''' , Nat. Bur. Standards  (1970)</TD></TR></table>

Revision as of 22:28, 21 November 2018

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)
How to Cite This Entry:
Euler numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_numbers&oldid=43465
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article