24 Bernoulli and Euler Polynomials Properties 24.1 Special Notation 24.3 Graphs

§24.2 Definitions and Generating Functions

ⓘ

Keywords:: Bernoulli numbers, definition, generating function
Referenced by:: §2.10(i)
Permalink:: http://dlmf.nist.gov/24.2
See also:: Annotations for Ch.24

§24.2(i) Bernoulli Numbers and Polynomials

ⓘ

Defines:: $B_{\NVar{n}}$ : Bernoulli numbers and $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials
Keywords:: Bernoulli polynomials, definitions, generating function
Notes:: See Nörlund (1924, pp. 18–23) or Milne-Thomson (1933, pp. 136–139).
Referenced by:: §13.8(iii), §24.1, §25.1, §25.2(iii), §26.14(iii), §3.5(iii), §4.19, §4.33, §5.11(i), §5.11(i), §5.15, §5.17
Permalink:: http://dlmf.nist.gov/24.2.i
See also:: Annotations for §24.2 and Ch.24

24.2.1		$\displaystyle\frac{t}{e^{t}-1}$	$\displaystyle=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!},$
$\|t\|<2\pi$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer and $t$ : real or complex Referenced by: §24.19(ii), §26.14(iii) Permalink: http://dlmf.nist.gov/24.2.E1 Encodings: TeX, pMML, png See also: Annotations for §24.2(i), §24.2 and Ch.24
24.2.2		$\displaystyle B_{2n+1}$	$\displaystyle=0$ ,
		$\displaystyle(-1)^{n+1}B_{2n}$	$\displaystyle>0$ ,
	$n=1,2,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers and $n$ : integer Referenced by: §2.10(i), (25.11.7) Permalink: http://dlmf.nist.gov/24.2.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §24.2(i), §24.2 and Ch.24
24.2.3		$\displaystyle\frac{te^{xt}}{e^{t}-1}$	$\displaystyle=\sum_{n=0}^{\infty}B_{n}\left(x\right)\frac{t^{n}}{n!},$
$\|t\|<2\pi$ .
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer, $t$ : real or complex and $x$ : real or complex A&S Ref: 23.1.1 Referenced by: §18.2(xii), §24.4(iv), §24.4(vii) Permalink: http://dlmf.nist.gov/24.2.E3 Encodings: TeX, pMML, png See also: Annotations for §24.2(i), §24.2 and Ch.24
24.2.4		$\displaystyle B_{n}$	$\displaystyle=B_{n}\left(0\right),$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer A&S Ref: 23.1.2 Referenced by: (25.11.7) Permalink: http://dlmf.nist.gov/24.2.E4 Encodings: TeX, pMML, png See also: Annotations for §24.2(i), §24.2 and Ch.24
24.2.5		$\displaystyle B_{n}\left(x\right)$	$\displaystyle=\sum_{k=0}^{n}{n\choose k}B_{k}x^{n-k}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.2.E5 Encodings: TeX, pMML, png See also: Annotations for §24.2(i), §24.2 and Ch.24

§24.2(ii) Euler Numbers and Polynomials

ⓘ

Defines:: $E_{\NVar{n}}$ : Euler numbers and $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials
Keywords:: Euler numbers, Euler polynomials, definition, generating function
Notes:: See Nörlund (1924, pp. 23–27) or Milne-Thomson (1933, pp. 146–148).
Referenced by:: §24.1, §4.19
Permalink:: http://dlmf.nist.gov/24.2.ii
See also:: Annotations for §24.2 and Ch.24

24.2.6	$\displaystyle\frac{2e^{t}}{e^{2t}+1}$	$\displaystyle=\sum_{n=0}^{\infty}E_{n}\frac{t^{n}}{n!},$
$\|t\|<\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer and $t$ : real or complex Permalink: http://dlmf.nist.gov/24.2.E6 Encodings: TeX, pMML, png See also: Annotations for §24.2(ii), §24.2 and Ch.24
24.2.7	$\displaystyle E_{2n+1}$	$\displaystyle=0$ ,
24.2.7	$\displaystyle(-1)^{n}E_{2n}$	$\displaystyle>0$ .
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers and $n$ : integer Referenced by: §24.9 Permalink: http://dlmf.nist.gov/24.2.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §24.2(ii), §24.2 and Ch.24
24.2.8	$\displaystyle\frac{2e^{xt}}{e^{t}+1}$	$\displaystyle=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!},$
$\|t\|<\pi$ ,
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer, $t$ : real or complex and $x$ : real or complex A&S Ref: 23.1.1 Referenced by: §18.2(xii), §24.4(iv), §24.4(vii) Permalink: http://dlmf.nist.gov/24.2.E8 Encodings: TeX, pMML, png See also: Annotations for §24.2(ii), §24.2 and Ch.24
24.2.9	$\displaystyle E_{n}$	$\displaystyle=2^{n}E_{n}\left(\tfrac{1}{2}\right)=\text{integer},$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials and $n$ : integer A&S Ref: 23.1.2 Permalink: http://dlmf.nist.gov/24.2.E9 Encodings: TeX, pMML, png See also: Annotations for §24.2(ii), §24.2 and Ch.24
24.2.10	$\displaystyle E_{n}\left(x\right)$	$\displaystyle=\sum_{k=0}^{n}{n\choose k}\frac{E_{k}}{2^{k}}(x-\tfrac{1}{2})^{n% -k}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.7 Permalink: http://dlmf.nist.gov/24.2.E10 Encodings: TeX, pMML, png See also: Annotations for §24.2(ii), §24.2 and Ch.24

§24.2(iii) Periodic Bernoulli and Euler Functions

ⓘ

Defines:: $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions and $\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Euler functions
Keywords:: periodic Bernoulli functions, periodic Euler functions
Referenced by:: §24.17(i), §24.17(ii), §25.11(iii), §25.2(iii)
Permalink:: http://dlmf.nist.gov/24.2.iii
See also:: Annotations for §24.2 and Ch.24

24.2.11		$\displaystyle\widetilde{B}_{n}\left(x\right)$	$\displaystyle=B_{n}\left(x\right)$ ,
		$\displaystyle\widetilde{E}_{n}\left(x\right)$	$\displaystyle=E_{n}\left(x\right)$ ,
	$0\leq x<1$ ,
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Euler functions, $n$ : integer and $x$ : real or complex Referenced by: (25.11.7) Permalink: http://dlmf.nist.gov/24.2.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §24.2(iii), §24.2 and Ch.24

24.2.12		$\displaystyle\widetilde{B}_{n}\left(x+1\right)$	$\displaystyle=\widetilde{B}_{n}\left(x\right),$
		$\displaystyle\widetilde{E}_{n}\left(x+1\right)$	$\displaystyle=-\widetilde{E}_{n}\left(x\right)$ ,
	$x\in\mathbb{R}$ .
ⓘ Symbols: $\in$ : element of, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Euler functions, $\mathbb{R}$ : real line, $n$ : integer and $x$ : real or complex Referenced by: (25.11.7) Permalink: http://dlmf.nist.gov/24.2.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §24.2(iii), §24.2 and Ch.24

§24.2(iv) Tables

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Notes:: The tables in this subsection are from Abramowitz and Stegun (1964, pp. 809–810).
Permalink:: http://dlmf.nist.gov/24.2.iv
See also:: Annotations for §24.2 and Ch.24

Table 24.2.1: Bernoulli and Euler numbers.

$n$	$B_{n}$	$E_{n}$
$0$	$1$	$1$
$1$	$-\tfrac{1}{2}$	$0$
$2$	$\tfrac{1}{6}$	$-1$
$4$	$-\tfrac{1}{30}$	$5$
$6$	$\tfrac{1}{42}$	$-61$
$8$	$-\tfrac{1}{30}$	$1385$
$10$	$\tfrac{5}{66}$	$-50521$
$12$	$-\tfrac{691}{2730}$	$27\;02765$
$14$	$\tfrac{7}{6}$	$-1993\;60981$
$16$	$-\tfrac{3617}{510}$	$1\;93915\;12145$

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers and $n$ : integer
Keywords:: Bernoulli numbers, Euler numbers, tables
Referenced by:: §24.1, (25.6.1)
Permalink:: http://dlmf.nist.gov/24.2.T1
See also:: Annotations for §24.2(iv), §24.2 and Ch.24

Table 24.2.2: Bernoulli and Euler polynomials.

$n$	$B_{n}\left(x\right)$	$E_{n}\left(x\right)$
$0$	$1$	$1$
$1$	$x-\frac{1}{2}$	$x-\frac{1}{2}$
$2$	$x^{2}-x+\frac{1}{6}$	$x^{2}-x$
$3$	$x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x$	$x^{3}-\frac{3}{2}x^{2}+\frac{1}{4}$
$4$	$x^{4}-2x^{3}+x^{2}-\frac{1}{30}$	$x^{4}-2x^{3}+x$
$5$	$x^{5}-\frac{5}{2}x^{4}+\frac{5}{3}x^{3}-\frac{1}{6}x$	$x^{5}-\frac{5}{2}x^{4}+\frac{5}{2}x^{2}-\frac{1}{2}$

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
Keywords:: Bernoulli polynomials, Euler polynomials, tables
Permalink:: http://dlmf.nist.gov/24.2.T2
See also:: Annotations for §24.2(iv), §24.2 and Ch.24

Table 24.2.3: Bernoulli numbers

B_{n}=N/D

$n$	$N$	$D$	$B_{n}$
$0$	$1$	$1$	$1.00000\;0000$
$1$	$-1$	$2$	$-5.00000\;0000$	$\times 10^{-1}$
$2$	$1$	$6$	$1.66666\;6667$	$\times 10^{-1}$
$4$	$-1$	$30$	$-3.33333\;3333$	$\times 10^{-2}$
$6$	$1$	$42$	$2.38095\;2381$	$\times 10^{-2}$
$8$	$-1$	$30$	$-3.33333\;3333$	$\times 10^{-2}$
$10$	$5$	$66$	$7.57575\;7576$	$\times 10^{-2}$
$12$	$-691$	$2730$	$-2.53113\;5531$	$\times 10^{-1}$
$14$	$7$	$6$	$1.16666\;6667$
$16$	$-3617$	$510$	$-7.09215\;6863$
$18$	$43867$	$798$	$5.49711\;7794$	$\times 10^{1}$
$20$	$-1\;74611$	$330$	$-5.29124\;2424$	$\times 10^{2}$
$22$	$8\;54513$	$138$	$6.19212\;3188$	$\times 10^{3}$
$24$	$-2363\;64091$	$2730$	$-8.65802\;5311$	$\times 10^{4}$
$26$	$85\;53103$	$6$	$1.42551\;7167$	$\times 10^{6}$
$28$	$-2\;37494\;61029$	$870$	$-2.72982\;3107$	$\times 10^{7}$
$30$	$861\;58412\;76005$	$14322$	$6.01580\;8739$	$\times 10^{8}$
$32$	$-770\;93210\;41217$	$510$	$-1.51163\;1577$	$\times 10^{10}$
$34$	$257\;76878\;58367$	$6$	$4.29614\;6431$	$\times 10^{11}$
$36$	$-26315\;27155\;30534\;77373$	$19\;19190$	$-1.37116\;5521$	$\times 10^{13}$
$38$	$2\;92999\;39138\;41559$	$6$	$4.88332\;3190$	$\times 10^{14}$
$40$	$-2\;61082\;71849\;64491\;22051$	$13530$	$-1.92965\;7934$	$\times 10^{16}$
$42$	$15\;20097\;64391\;80708\;02691$	$1806$	$8.41693\;0476$	$\times 10^{17}$
$44$	$-278\;33269\;57930\;10242\;35023$	$690$	$-4.03380\;7185$	$\times 10^{19}$
$46$	$5964\;51111\;59391\;21632\;77961$	$282$	$2.11507\;4864$	$\times 10^{21}$
$48$	$-560\;94033\;68997\;81768\;62491\;27547$	$46410$	$-1.20866\;2652$	$\times 10^{23}$
$50$	$49\;50572\;05241\;07964\;82124\;77525$	$66$	$7.50086\;6746$	$\times 10^{24}$
$52$	$-80116\;57181\;35489\;95734\;79249\;91853$	$1590$	$-5.03877\;8101$	$\times 10^{26}$
$54$	$29\;14996\;36348\;84862\;42141\;81238\;12691$	$798$	$3.65287\;7648$	$\times 10^{28}$
$56$	$-2479\;39292\;93132\;26753\;68541\;57396\;63229$	$870$	$-2.84987\;6930$	$\times 10^{30}$
$58$	$84483\;61334\;88800\;41862\;04677\;59940\;36021$	$354$	$2.38654\;2750$	$\times 10^{32}$
$60$	$-121\;52331\;40483\;75557\;20403\;04994\;07982\;02460\;41491$	$567\;86730$	$-2.13999\;4926$	$\times 10^{34}$

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.2.T3
See also:: Annotations for §24.2(iv), §24.2 and Ch.24

Table 24.2.4: Euler numbers

E_{n}

$n$	$E_{n}$
$0$	$1$
$2$	$-1$
$4$	$5$
$6$	$-61$
$8$	$1385$
$10$	$-50521$
$12$	$27\;02765$
$14$	$-1993\;60981$
$16$	$1\;93915\;12145$
$18$	$-240\;48796\;75441$
$20$	$37037\;11882\;37525$
$22$	$-69\;34887\;43931\;37901$
$24$	$15514\;53416\;35570\;86905$
$26$	$-40\;87072\;50929\;31238\;92361$
$28$	$12522\;59641\;40362\;98654\;68285$
$30$	$-44\;15438\;93249\;02310\;45536\;82821$
$32$	$17751\;93915\;79539\;28943\;66647\;89665$
$34$	$-80\;72329\;92358\;87898\;06216\;82474\;53281$
$36$	$41222\;06033\;95177\;02122\;34707\;96712\;59045$
$38$	$-234\;89580\;52704\;31082\;52017\;82857\;61989\;47741$
$40$	$1\;48511\;50718\;11498\;00178\;77156\;78140\;58266\;84425$
$42$	$-1036\;46227\;33519\;61211\;93979\;57304\;74518\;59763\;10201$
$44$	$7\;94757\;94225\;97592\;70360\;80405\;10088\;07061\;95192\;73805$
$46$	$-6667\;53751\;66855\;44977\;43502\;84747\;73748\;19752\;41076\;84661$
$48$	$60\;96278\;64556\;85421\;58691\;68574\;28768\;43153\;97653\;90444\;35185$
$50$	$-60532\;85248\;18862\;18963\;14383\;78511\;16490\;88103\;49822\;51468\;15121$
$52$	$650\;61624\;86684\;60884\;77158\;70634\;08082\;29834\;83644\;23676\;53855\;76565$
$54$	$-7\;54665\;99390\;08739\;09806\;14325\;65889\;73674\;42122\;40024\;71169\;9858% 6\;45581$
$56$	$9420\;32189\;64202\;41204\;20228\;62376\;90583\;22720\;93888\;52599\;64600\;93% 949\;05945$
$58$	$-126\;22019\;25180\;62187\;19903\;40923\;72874\;89255\;48234\;10611\;91825\;59% 406\;99649\;20041$
$60$	$1\;81089\;11496\;57923\;04965\;45807\;74165\;21586\;88733\;48734\;92363\;14106% \;00809\;54542\;31325$

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.2.T4
See also:: Annotations for §24.2(iv), §24.2 and Ch.24

Table 24.2.5: Coefficients

b_{n,k}

of the Bernoulli polynomials

B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}

	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
$0$	$1$
$1$	$-\tfrac{1}{2}$	$1$
$2$	$\tfrac{1}{6}$	$-1$	$1$
$3$	$0$	$\tfrac{1}{2}$	$-\tfrac{3}{2}$	$1$
$4$	$-\tfrac{1}{30}$	$0$	$1$	$-2$	$1$
$5$	$0$	$-\tfrac{1}{6}$	$0$	$\tfrac{5}{3}$	$-\tfrac{5}{2}$	$1$
$6$	$\tfrac{1}{42}$	$0$	$-\tfrac{1}{2}$	$0$	$\tfrac{5}{2}$	$-3$	$1$
$7$	$0$	$\tfrac{1}{6}$	$0$	$-\tfrac{7}{6}$	$0$	$\tfrac{7}{2}$	$-\tfrac{7}{2}$	$1$
$8$	$-\tfrac{1}{30}$	$0$	$\tfrac{2}{3}$	$0$	$-\tfrac{7}{3}$	$0$	$\tfrac{14}{3}$	$-4$	$1$
$9$	$0$	$-\tfrac{3}{10}$	$0$	$2$	$0$	$-\tfrac{21}{5}$	$0$	$6$	$-\tfrac{9}{2}$	$1$
$10$	$\tfrac{5}{66}$	$0$	$-\tfrac{3}{2}$	$0$	$5$	$0$	$-7$	$0$	$\tfrac{15}{2}$	$-5$	$1$
$11$	$0$	$\tfrac{5}{6}$	$0$	$-\tfrac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\tfrac{55}{6}$	$-\tfrac{11}{2}$	$1$
$12$	$-\tfrac{691}{2730}$	$0$	$5$	$0$	$-\tfrac{33}{2}$	$0$	$22$	$0$	$-\tfrac{33}{2}$	$0$	$11$	$-6$	$1$
$13$	$0$	$-\tfrac{691}{210}$	$0$	$\tfrac{65}{3}$	$0$	$-\tfrac{429}{10}$	$0$	$\tfrac{286}{7}$	$0$	$-\tfrac{143}{6}$	$0$	$13$	$-\tfrac{13}{2}$	$1$
$14$	$\tfrac{7}{6}$	$0$	$-\tfrac{691}{30}$	$0$	$\tfrac{455}{6}$	$0$	$-\tfrac{1001}{10}$	$0$	$\tfrac{143}{2}$	$0$	$-\tfrac{1001}{30}$	$0$	$\tfrac{91}{6}$	$-7$	$1$
$15$	$0$	$\tfrac{35}{2}$	$0$	$-\tfrac{691}{6}$	$0$	$\tfrac{455}{2}$	$0$	$-\tfrac{429}{2}$	$0$	$\tfrac{715}{6}$	$0$	$-\tfrac{91}{2}$	$0$	$\tfrac{35}{2}$	$-\tfrac{15}{2}$	$1$

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.T5
See also:: Annotations for §24.2(iv), §24.2 and Ch.24

Table 24.2.6: Coefficients

e_{n,k}

of the Euler polynomials

E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}

	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
$0$	$1$
$1$	$-\tfrac{1}{2}$	$1$
$2$	$0$	$-1$	$1$
$3$	$\tfrac{1}{4}$	$0$	$-\tfrac{3}{2}$	$1$
$4$	$0$	$1$	0	$-2$	$1$
$5$	$-\tfrac{1}{2}$	$0$	$\tfrac{5}{2}$	$0$	$-\tfrac{5}{2}$	$1$
$6$	$0$	$-3$	$0$	$5$	$0$	$-3$	$1$
$7$	$\tfrac{17}{8}$	$0$	$-\tfrac{21}{2}$	$0$	$\tfrac{35}{4}$	$0$	$-\tfrac{7}{2}$	$1$
$8$	$0$	$17$	$0$	$-28$	$0$	$14$	$0$	$-4$	$1$
$9$	$-\tfrac{31}{2}$	$0$	$\tfrac{153}{2}$	$0$	$-63$	$0$	$21$	$0$	$-\tfrac{9}{2}$	$1$
$10$	$0$	$-155$	$0$	$255$	$0$	$-126$	$0$	$30$	$0$	$-5$	$1$
$11$	$\tfrac{691}{4}$	$0$	$-\tfrac{1705}{2}$	$0$	$\tfrac{2805}{4}$	$0$	$-231$	$0$	$\tfrac{165}{4}$	$0$	$-\tfrac{11}{2}$	$1$
$12$	$0$	$2073$	$0$	$-3410$	$0$	$1683$	$0$	$-396$	$0$	$55$	$0$	$-6$	$1$
$13$	$-\tfrac{5461}{2}$	$0$	$\tfrac{26949}{2}$	$0$	$-\tfrac{22165}{2}$	$0$	$\tfrac{7293}{2}$	$0$	$-\tfrac{1287}{2}$	$0$	$\tfrac{143}{2}$	$0$	$-\tfrac{13}{2}$	$1$
$14$	$0$	$-38227$	$0$	$62881$	$0$	$-31031$	$0$	$7293$	$0$	$-1001$	$0$	$91$	$0$	$-7$	$1$
$15$	$\tfrac{929569}{16}$	$0$	$-\tfrac{573405}{2}$	$0$	$\tfrac{943215}{4}$	$0$	$-\tfrac{155155}{2}$	$0$	$\tfrac{109395}{8}$	$0$	$-\tfrac{3003}{2}$	$0$	$\tfrac{455}{4}$	$0$	$-\tfrac{15}{2}$	$1$

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.T6
See also:: Annotations for §24.2(iv), §24.2 and Ch.24

24.1 Special Notation 24.3 Graphs

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§24.2 Definitions and Generating Functions

Contents

§24.2(i) Bernoulli Numbers and Polynomials

§24.2(ii) Euler Numbers and Polynomials

§24.2(iii) Periodic Bernoulli and Euler Functions

§24.2(iv) Tables