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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 | + | {{TEX|done}} |
| + | The coefficients $E_n$ in the expansion |
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− | <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>
| + | $$\frac1{\cosh z}=\sum_{n=0}^\infty E_n\frac{z^n}{n!}.$$ |
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− | 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 | + | The recurrence formula for the Euler numbers ($E^n\equiv E_n$ in symbolic notation) has the form |
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− | <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>
| + | $$(E+1)^n+(E-1)^n=0,\quad E_0=1.$$ |
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− | 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 | + | 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 |
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− | <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>
| + | $$E_{n-1}=\frac{(4B-1)^n-(4B-3)^n}{2n},$$ |
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− | <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>
| + | $$E_{2n}=\frac{4^{2n+1}}{2n+1}\left(B-\frac14\right)^{2n+1}.$$ |
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| The Euler numbers are used in the summation of series. For example, | | The Euler numbers are used in the summation of series. For example, |
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− | <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>
| + | $$\sum_{k=0}^\infty(-1)^k\frac1{(2k+1)^{2n+1}}=\frac{\pi^{2n+1}}{2^{2n+2}(2n)!}|E_{2n}|.$$ |
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− | Sometimes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036540/e03654018.png" /> are called the Euler numbers. | + | Sometimes the $|E_{2n}|$ are called the Euler numbers. |
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| 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" />. | + | 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 $E_n=2^nE_n(1/2)$. |
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| ====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) |
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=12473
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article