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24 Bernoulli and Euler PolynomialsProperties

§24.2 Definitions and Generating Functions

Contents

§24.2(i) Bernoulli Numbers and Polynomials

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

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(x-12)n-k.

See also (4.19.5).

§24.2(iii) Periodic Bernoulli and Euler Functions

§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 x-12 x-12
2 x2-x+16 x2-x
3 x3-32x2+12x x3-32x2+14
4 x4-2x3+x2-130 x4-2x3+x
5 x5-52x4+53x3-16x x5-52x4+52x2-12
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