§24.2 Definitions and Generating Functions

Keywords:: Bernoulli numbers
Referenced by:: §2.10(i)
Permalink:: http://dlmf.nist.gov/24.2

§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

§24.2(i) Bernoulli Numbers and Polynomials

Notes:: See Nörlund (1924, pp. 18–23) or Milne-Thomson (1933, pp. 136–139).
Defines:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials
Keywords:: Bernoulli polynomials
Referenced by:: ¶ ‣ §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

24.2.1 $\frac{t}{e^{t}-1}=\sum _{{n=0}}^{{\infty}}\mathop{B_{{n}}\/}\nolimits\frac{t^{n}}{n!},$ $|t|<2\pi$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : 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

24.2.2

$\mathop{B_{{2n+1}}\/}\nolimits=0$ ,

$(-1)^{{n+1}}\mathop{B_{{2n}}\/}\nolimits>0$ , $n=1,2,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials and $n$ : integer
Referenced by:: ¶ ‣ §2.10(i)
Permalink:: http://dlmf.nist.gov/24.2.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

24.2.3 $\frac{te^{{xt}}}{e^{t}-1}=\sum _{{n=0}}^{{\infty}}\mathop{B_{{n}}\/}\nolimits\!\left(x\right)\frac{t^{n}}{n!},$ $|t|<2\pi$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : integer, $t$ : real or complex and $x$ : real or complex
A&S Ref:: 23.1.1
Referenced by:: §24.4(iv), §24.4(vii)
Permalink:: http://dlmf.nist.gov/24.2.E3
Encodings:: TeX, pMML, png

24.2.4 $\mathop{B_{{n}}\/}\nolimits=\mathop{B_{{n}}\/}\nolimits\!\left(0\right),$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: TeX, pMML, png

24.2.5 $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{k=0}}^{n}{n\choose k}\mathop{B_{{k}}\/}\nolimits x^{{n-k}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.E5
Encodings:: TeX, pMML, png

§24.2(ii) Euler Numbers and Polynomials

Notes:: See Nörlund (1924, pp. 23–27) or Milne-Thomson (1933, pp. 146–148).
Defines:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials
Keywords:: Euler numbers, Euler polynomials
Referenced by:: ¶ ‣ §24.1, §4.19
Permalink:: http://dlmf.nist.gov/24.2.ii

24.2.6 $\frac{2e^{t}}{e^{{2t}}+1}=\sum _{{n=0}}^{{\infty}}\mathop{E_{{n}}\/}\nolimits\frac{t^{n}}{n!},$ $|t|<\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.E6
Encodings:: TeX, pMML, png

24.2.7

$\mathop{E_{{2n+1}}\/}\nolimits=0$ ,

$(-1)^{n}\mathop{E_{{2n}}\/}\nolimits>0$ .

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.2.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

24.2.8 $\frac{2e^{{xt}}}{e^{t}+1}=\sum _{{n=0}}^{{\infty}}\mathop{E_{{n}}\/}\nolimits\!\left(x\right)\frac{t^{n}}{n!},$ $|t|<\pi$ ,

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : integer, $t$ : real or complex and $x$ : real or complex
A&S Ref:: 23.1.1
Referenced by:: §24.4(iv), §24.4(vii)
Permalink:: http://dlmf.nist.gov/24.2.E8
Encodings:: TeX, pMML, png

24.2.9 $\mathop{E_{{n}}\/}\nolimits=2^{n}\mathop{E_{{n}}\/}\nolimits\!\left(\tfrac{1}{2}\right)=\text{integer},$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.2
Permalink:: http://dlmf.nist.gov/24.2.E9
Encodings:: TeX, pMML, png

24.2.10 $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{k=0}}^{n}{n\choose k}\frac{\mathop{E_{{k}}\/}\nolimits}{2^{k}}(x-\tfrac{1}{2})^{{n-k}}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : 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

§24.2(iii) Periodic Bernoulli and Euler Functions

Defines:: $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions and $\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

24.2.11

$\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ ,

$\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ , $0\leq x<1$ ,

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Euler functions, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

24.2.12

$\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x+1\right)=\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right),$

$\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x+1\right)=-\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)$ , $x\in\Real$ .

Symbols:: $\in$ : element of, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Euler functions, $\Real$ : real line (excluding infinity), $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

§24.2(iv) Tables

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

Table 24.2.1: Bernoulli and Euler numbers.

$n$	$\mathop{B_{{n}}\/}\nolimits$	$\mathop{E_{{n}}\/}\nolimits$
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:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
Keywords:: Bernoulli numbers, Euler numbers
Referenced by:: ¶ ‣ §24.1
Permalink:: http://dlmf.nist.gov/24.2.T1

Table 24.2.2: Bernoulli and Euler polynomials.

$n$	$\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$	$\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$
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:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
Keywords:: Bernoulli polynomials, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.2.T2

Table 24.2.3: Bernoulli numbers $\mathop{B_{{n}}\/}\nolimits=N/D$ .

$n$	$N$	$D$	$\mathop{B_{{n}}\/}\nolimits$
0	1	1	1.00000 0000
1	−1	2	−5.00000 0000	×10⁻¹
2	1	6	1.66666 6667	×10⁻¹
4	−1	30	−3.33333 3333	×10⁻²
6	1	42	2.38095 2381	×10⁻²
8	−1	30	−3.33333 3333	×10⁻²
10	5	66	7.57575 7576	×10⁻²
12	−691	2730	−2.53113 5531	×10⁻¹
14	7	6	1.16666 6667
16	−3617	510	−7.09215 6863
18	43867	798	5.49711 7794	×10¹
20	−1 74611	330	−5.29124 2424	×10²
22	8 54513	138	6.19212 3188	×10³
24	−2363 64091	2730	−8.65802 5311	×10⁴
26	85 53103	6	1.42551 7167	×10⁶
28	−2 37494 61029	870	−2.72982 3107	×10⁷
30	861 58412 76005	14322	6.01580 8739	×10⁸
32	−770 93210 41217	510	−1.51163 1577	×10¹⁰
34	257 76878 58367	6	4.29614 6431	×10¹¹
36	−26315 27155 30534 77373	19 19190	−1.37116 5521	×10¹³
38	2 92999 39138 41559	6	4.88332 3190	×10¹⁴
40	−2 61082 71849 64491 22051	13530	−1.92965 7934	×10¹⁶
42	15 20097 64391 80708 02691	1806	8.41693 0476	×10¹⁷
44	−278 33269 57930 10242 35023	690	−4.03380 7185	×10¹⁹
46	5964 51111 59391 21632 77961	282	2.11507 4864	×10²¹
48	−560 94033 68997 81768 62491 27547	46410	−1.20866 2652	×10²³
50	49 50572 05241 07964 82124 77525	66	7.50086 6746	×10²⁴
52	−80116 57181 35489 95734 79249 91853	1590	−5.03877 8101	×10²⁶
54	29 14996 36348 84862 42141 81238 12691	798	3.65287 7648	×10²⁸
56	−2479 39292 93132 26753 68541 57396 63229	870	−2.84987 6930	×10³⁰
58	84483 61334 88800 41862 04677 59940 36021	354	2.38654 2750	×10³²
60	−121 52331 40483 75557 20403 04994 07982 02460 41491	567 86730	−2.13999 4926	×10³⁴

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.2.T3

Table 24.2.4: Euler numbers $\mathop{E_{{n}}\/}\nolimits$ .

$n$	$\mathop{E_{{n}}\/}\nolimits$
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 98586 45581
56	9420 32189 64202 41204 20228 62376 90583 22720 93888 52599 64600 93949 05945
58	−126 22019 25180 62187 19903 40923 72874 89255 48234 10611 91825 59406 99649 20041
60	1 81089 11496 57923 04965 45807 74165 21586 88733 48734 92363 14106 00809 54542 31325

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.2.T4

Table 24.2.5: Coefficients $b_{{n,k}}$ of the Bernoulli polynomials $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{k=0}}^{{n}}b_{{n,k}}x^{k}$ .

	$k$
$n$	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:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.T5

Table 24.2.6: Coefficients $e_{{n,k}}$ of the Euler polynomials $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{k=0}}^{{n}}e_{{n,k}}x^{k}$ .

	$k$
$n$	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