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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
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
§24.2(i)
Bernoulli Numbers and Polynomials
ⓘ
Defines:
B
n
: Bernoulli numbers
and
B
n
(
x
)
: 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
t
e
t
−
1
=
∑
n
=
0
∞
B
n
t
n
n
!
,
|
t
|
<
2
π
.
ⓘ
Symbols:
B
n
: Bernoulli numbers
,
π
: the ratio of the circumference of a circle to its diameter
,
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
B
2
n
+
1
=
0
,
(
−
1
)
n
+
1
B
2
n
>
0
,
n
=
1
,
2
,
…
.
ⓘ
Symbols:
B
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
t
e
x
t
e
t
−
1
=
∑
n
=
0
∞
B
n
(
x
)
t
n
n
!
,
|
t
|
<
2
π
.
ⓘ
Symbols:
B
n
(
x
)
: Bernoulli polynomials
,
π
: the ratio of the circumference of a circle to its diameter
,
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
B
n
=
B
n
(
0
)
,
ⓘ
Symbols:
B
n
: Bernoulli numbers
,
B
n
(
x
)
: 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
B
n
(
x
)
=
∑
k
=
0
n
(
n
k
)
B
k
x
n
−
k
.
ⓘ
Symbols:
B
n
: Bernoulli numbers
,
B
n
(
x
)
: Bernoulli polynomials
,
(
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
See also:
Annotations for
§24.2(i)
,
§24.2
and
Ch.24
See also §§
4.19
and
4.33
.
§24.2(ii)
Euler Numbers and Polynomials
ⓘ
Defines:
E
n
: Euler numbers
and
E
n
(
x
)
: 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
2
e
t
e
2
t
+
1
=
∑
n
=
0
∞
E
n
t
n
n
!
,
|
t
|
<
1
2
π
,
ⓘ
Symbols:
E
n
: Euler numbers
,
π
: the ratio of the circumference of a circle to its diameter
,
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
E
2
n
+
1
=
0
,
(
−
1
)
n
E
2
n
>
0
.
ⓘ
Symbols:
E
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
2
e
x
t
e
t
+
1
=
∑
n
=
0
∞
E
n
(
x
)
t
n
n
!
,
|
t
|
<
π
,
ⓘ
Symbols:
E
n
(
x
)
: Euler polynomials
,
π
: the ratio of the circumference of a circle to its diameter
,
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
E
n
=
2
n
E
n
(
1
2
)
=
integer
,
ⓘ
Symbols:
E
n
: Euler numbers
,
E
n
(
x
)
: 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
E
n
(
x
)
=
∑
k
=
0
n
(
n
k
)
E
k
2
k
(
x
−
1
2
)
n
−
k
.
ⓘ
Symbols:
E
n
: Euler numbers
,
E
n
(
x
)
: Euler polynomials
,
(
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
See also:
Annotations for
§24.2(ii)
,
§24.2
and
Ch.24
See also (
4.19.5
).
§24.2(iii)
Periodic Bernoulli and Euler Functions
ⓘ
Defines:
B
~
n
(
x
)
: periodic Bernoulli functions
and
E
~
n
(
x
)
: 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
B
~
n
(
x
)
=
B
n
(
x
)
,
E
~
n
(
x
)
=
E
n
(
x
)
,
0
≤
x
<
1
,
ⓘ
Symbols:
B
n
(
x
)
: Bernoulli polynomials
,
E
n
(
x
)
: Euler polynomials
,
B
~
n
(
x
)
: periodic Bernoulli functions
,
E
~
n
(
x
)
: 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
B
~
n
(
x
+
1
)
=
B
~
n
(
x
)
,
E
~
n
(
x
+
1
)
=
−
E
~
n
(
x
)
,
x
∈
ℝ
.
ⓘ
Symbols:
∈
: element of
,
B
~
n
(
x
)
: periodic Bernoulli functions
,
E
~
n
(
x
)
: periodic Euler functions
,
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
ⓘ
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
−
1
2
0
2
1
6
−
1
4
−
1
30
5
6
1
42
−
61
8
−
1
30
1385
10
5
66
−
50521
12
−
691
2730
27 02765
14
7
6
−
1993 60981
16
−
3617
510
1 93915 12145
ⓘ
Symbols:
B
n
: Bernoulli numbers
,
E
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
(
x
)
E
n
(
x
)
0
1
1
1
x
−
1
2
x
−
1
2
2
x
2
−
x
+
1
6
x
2
−
x
3
x
3
−
3
2
x
2
+
1
2
x
x
3
−
3
2
x
2
+
1
4
4
x
4
−
2
x
3
+
x
2
−
1
30
x
4
−
2
x
3
+
x
5
x
5
−
5
2
x
4
+
5
3
x
3
−
1
6
x
x
5
−
5
2
x
4
+
5
2
x
2
−
1
2
ⓘ
Symbols:
B
n
(
x
)
: Bernoulli polynomials
,
E
n
(
x
)
: 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
×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:
B
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 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:
E
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
(
x
)
=
∑
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
−
1
2
1
2
1
6
−
1
1
3
0
1
2
−
3
2
1
4
−
1
30
0
1
−
2
1
5
0
−
1
6
0
5
3
−
5
2
1
6
1
42
0
−
1
2
0
5
2
−
3
1
7
0
1
6
0
−
7
6
0
7
2
−
7
2
1
8
−
1
30
0
2
3
0
−
7
3
0
14
3
−
4
1
9
0
−
3
10
0
2
0
−
21
5
0
6
−
9
2
1
10
5
66
0
−
3
2
0
5
0
−
7
0
15
2
−
5
1
11
0
5
6
0
−
11
2
0
11
0
−
11
0
55
6
−
11
2
1
12
−
691
2730
0
5
0
−
33
2
0
22
0
−
33
2
0
11
−
6
1
13
0
−
691
210
0
65
3
0
−
429
10
0
286
7
0
−
143
6
0
13
−
13
2
1
14
7
6
0
−
691
30
0
455
6
0
−
1001
10
0
143
2
0
−
1001
30
0
91
6
−
7
1
15
0
35
2
0
−
691
6
0
455
2
0
−
429
2
0
715
6
0
−
91
2
0
35
2
−
15
2
1
ⓘ
Symbols:
B
n
(
x
)
: 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
(
x
)
=
∑
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
−
1
2
1
2
0
−
1
1
3
1
4
0
−
3
2
1
4
0
1
0
−
2
1
5
−
1
2
0
5
2
0
−
5
2
1
6
0
−
3
0
5
0
−
3
1
7
17
8
0
−
21
2
0
35
4
0
−
7
2
1
8
0
17
0
−
28
0
14
0
−
4
1
9
−
31
2
0
153
2
0
−
63
0
21
0
−
9
2
1
10
0
−
155
0
255
0
−
126
0
30
0
−
5
1
11
691
4
0
−
1705
2
0
2805
4
0
−
231
0
165
4
0
−
11
2
1
12
0
2073
0
−
3410
0
1683
0
−
396
0
55
0
−
6
1
13
−
5461
2
0
26949
2
0
−
22165
2
0
7293
2
0
−
1287
2
0
143
2
0
−
13
2
1
14
0
−
38227
0
62881
0
−
31031
0
7293
0
−
1001
0
91
0
−
7
1
15
929569
16
0
−
573405
2
0
943215
4
0
−
155155
2
0
109395
8
0
−
3003
2
0
455
4
0
−
15
2
1
ⓘ
Symbols:
E
n
(
x
)
: 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