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24 Bernoulli and Euler PolynomialsProperties

§24.2 Definitions and Generating Functions

Contents
  1. §24.2(i) Bernoulli Numbers and Polynomials
  2. §24.2(ii) Euler Numbers and Polynomials
  3. §24.2(iii) Periodic Bernoulli and Euler Functions
  4. §24.2(iv) Tables

§24.2(i) Bernoulli Numbers and Polynomials

24.2.1 tet1 =n=0Bntnn!,
|t|<2π.
24.2.2 B2n+1 =0,
(1)n+1B2n >0,
n=1,2,.
24.2.3 textet1 =n=0Bn(x)tnn!,
|t|<2π.
24.2.4 Bn =Bn(0),
24.2.5 Bn(x) =k=0n(nk)Bkxnk.

See also §§4.19 and 4.33.

§24.2(ii) Euler Numbers and Polynomials

24.2.6 2ete2t+1 =n=0Entnn!,
|t|<12π,
24.2.7 E2n+1 =0,
(1)nE2n >0.
24.2.8 2extet+1 =n=0En(x)tnn!,
|t|<π,
24.2.9 En =2nEn(12)=integer,
24.2.10 En(x) =k=0n(nk)Ek2k(x12)nk.

See also (4.19.5).

§24.2(iii) Periodic Bernoulli and Euler Functions

24.2.11 B~n(x) =Bn(x),
E~n(x) =En(x),
0x<1,
24.2.12 B~n(x+1) =B~n(x),
E~n(x+1) =E~n(x),
x.

§24.2(iv) Tables

Table 24.2.1: Bernoulli and Euler numbers.
n Bn En
0 1 1
1 12 0
2 16 1
4 130 5
6 142 61
8 130 1385
10 566 50521
12 6912730 27 02765
14 76 1993 60981
16 3617510 1 93915 12145
Table 24.2.2: Bernoulli and Euler polynomials.
n Bn(x) En(x)
0 1 1
1 x12 x12
2 x2x+16 x2x
3 x332x2+12x x332x2+14
4 x42x3+x2130 x42x3+x
5 x552x4+53x316x x552x4+52x212
Table 24.2.3: Bernoulli numbers Bn=N/D.
n N D Bn
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
Table 24.2.4: Euler numbers En.
n En
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
Table 24.2.5: Coefficients bn,k of the Bernoulli polynomials Bn(x)=k=0nbn,kxk.
k
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1
1 12 1
2 16 1 1
3 0 12 32 1
4 130 0 1 2 1
5 0 16 0 53 52 1
6 142 0 12 0 52 3 1
7 0 16 0 76 0 72 72 1
8 130 0 23 0 73 0 143 4 1
9 0 310 0 2 0 215 0 6 92 1
10 566 0 32 0 5 0 7 0 152 5 1
11 0 56 0 112 0 11 0 11 0 556 112 1
12 6912730 0 5 0 332 0 22 0 332 0 11 6 1
13 0 691210 0 653 0 42910 0 2867 0 1436 0 13 132 1
14 76 0 69130 0 4556 0 100110 0 1432 0 100130 0 916 7 1
15 0 352 0 6916 0 4552 0 4292 0 7156 0 912 0 352 152 1
Table 24.2.6: Coefficients en,k of the Euler polynomials En(x)=k=0nen,kxk.
k
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1
1 12 1
2 0 1 1
3 14 0 32 1
4 0 1 0 2 1
5 12 0 52 0 52 1
6 0 3 0 5 0 3 1
7 178 0 212 0 354 0 72 1
8 0 17 0 28 0 14 0 4 1
9 312 0 1532 0 63 0 21 0 92 1
10 0 155 0 255 0 126 0 30 0 5 1
11 6914 0 17052 0 28054 0 231 0 1654 0 112 1
12 0 2073 0 3410 0 1683 0 396 0 55 0 6 1
13 54612 0 269492 0 221652 0 72932 0 12872 0 1432 0 132 1
14 0 38227 0 62881 0 31031 0 7293 0 1001 0 91 0 7 1
15 92956916 0 5734052 0 9432154 0 1551552 0 1093958 0 30032 0 4554 0 152 1