Bernoulli numbers and polynomials

Symbol: BernoulliB —

B_{n}

— Bernoulli number

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0}$	$B_{n} \in \mathbb{Q}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \;\implies\; \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("ac8eca"),
    SymbolDefinition(BernoulliB, BernoulliB(n), "Bernoulli number"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(Element(n, ZZGreaterEqual(0)), Element(BernoulliB(n), QQ)))))

1f88a4

Symbol: BernoulliPolynomial —

B_{n}\!\left(z\right)

— Bernoulli polynomial

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{R}$	$B_{n}\!\left(z\right) \in \mathbb{R}$
$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$	$B_{n}\!\left(z\right) \in \mathbb{C}$
$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in R \;\mathbin{\operatorname{and}}\; R \in \operatorname{Rings} \;\mathbin{\operatorname{and}}\; \mathbb{Q} \subseteq R$	$B_{n}\!\left(z\right) \in R$

Table data:

\left(P, Q\right)

such that

\left(P\right) \;\implies\; \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("1f88a4"),
    SymbolDefinition(BernoulliPolynomial, BernoulliPolynomial(n, z), "Bernoulli polynomial"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(And(Element(n, ZZGreaterEqual(0)), Element(z, RR)), Element(BernoulliPolynomial(n, z), RR)), Tuple(And(Element(n, ZZGreaterEqual(0)), Element(z, CC)), Element(BernoulliPolynomial(n, z), CC)), Tuple(And(Element(n, ZZGreaterEqual(0)), Element(z, R), Element(R, Rings), SubsetEqual(QQ, R)), Element(BernoulliPolynomial(n, z), R)))))

b6111c

B_{n} = \frac{\text{A027641}\!\left(n\right)}{\text{A027642}\!\left(n\right)}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

References:

Sequence A027641 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)
Sequence A027642 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

B_{n} = \frac{\text{A027641}\!\left(n\right)}{\text{A027642}\!\left(n\right)}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b6111c"),
    Formula(Equal(BernoulliB(n), Div(SloaneA("A027641", n), SloaneA("A027642", n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

aed6bd

Table of

B_{n}

for

0 \le n \le 50

$n$	$B_{n}$	$B_{n} \; (\text{nearest } 30 \text{D})$
0	1	1.00000000000000000000000000000
1	-1/2	-0.500000000000000000000000000000
2	1/6	0.166666666666666666666666666667
3	0	0
4	-1/30	-0.0333333333333333333333333333333
5	0	0
6	1/42	0.0238095238095238095238095238095
7	0	0
8	-1/30	-0.0333333333333333333333333333333
9	0	0
10	5/66	0.0757575757575757575757575757576
11	0	0
12	-691/2730	-0.253113553113553113553113553114
13	0	0
14	7/6	1.16666666666666666666666666667
15	0	0
16	-3617/510	-7.09215686274509803921568627451
17	0	0
18	43867/798	54.9711779448621553884711779449
19	0	0
20	-174611/330	-529.124242424242424242424242424
21	0	0
22	854513/138	6192.12318840579710144927536232
23	0	0
24	-236364091/2730	-86580.2531135531135531135531136
25	0	0
26	8553103/6	1425517.16666666666666666666667
27	0	0
28	-23749461029/870	-27298231.0678160919540229885057
29	0	0
30	8615841276005/14322	601580873.900642368384303868175
31	0	0
32	-7709321041217/510	-15116315767.0921568627450980392
33	0	0
34	2577687858367/6	429614643061.166666666666666667
35	0	0
36	-26315271553053477373/1919190	-13711655205088.3327721590879486
37	0	0
38	2929993913841559/6	488332318973593.166666666666667
39	0	0
40	-261082718496449122051/13530	-19296579341940068.1486326681449
41	0	0
42	1520097643918070802691/1806	841693047573682615.000553709856
43	0	0
44	-27833269579301024235023/690	-40338071854059455413.0768115942
45	0	0
46	596451111593912163277961/282	2115074863808199160560.14539007
47	0	0
48	-5609403368997817686249127547/46410	-120866265222965259346027.311937
49	0	0
50	495057205241079648212477525/66	7500866746076964366855720.07576

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number

Source code for this entry:

Entry(ID("aed6bd"),
    Description("Table of", BernoulliB(n), "for", LessEqual(0, n, 50)),
    Table(Var(n), TableValueHeadings(n, BernoulliB(n), NearestDecimal(BernoulliB(n), 30)), TableSplit(1), List(Tuple(0, 1, Decimal("1.00000000000000000000000000000")), Tuple(1, Neg(Div(1, 2)), Decimal("-0.500000000000000000000000000000")), Tuple(2, Div(1, 6), Decimal("0.166666666666666666666666666667")), Tuple(3, 0, Decimal("0")), Tuple(4, Neg(Div(1, 30)), Decimal("-0.0333333333333333333333333333333")), Tuple(5, 0, Decimal("0")), Tuple(6, Div(1, 42), Decimal("0.0238095238095238095238095238095")), Tuple(7, 0, Decimal("0")), Tuple(8, Neg(Div(1, 30)), Decimal("-0.0333333333333333333333333333333")), Tuple(9, 0, Decimal("0")), Tuple(10, Div(5, 66), Decimal("0.0757575757575757575757575757576")), Tuple(11, 0, Decimal("0")), Tuple(12, Neg(Div(691, 2730)), Decimal("-0.253113553113553113553113553114")), Tuple(13, 0, Decimal("0")), Tuple(14, Div(7, 6), Decimal("1.16666666666666666666666666667")), Tuple(15, 0, Decimal("0")), Tuple(16, Neg(Div(3617, 510)), Decimal("-7.09215686274509803921568627451")), Tuple(17, 0, Decimal("0")), Tuple(18, Div(43867, 798), Decimal("54.9711779448621553884711779449")), Tuple(19, 0, Decimal("0")), Tuple(20, Neg(Div(174611, 330)), Decimal("-529.124242424242424242424242424")), Tuple(21, 0, Decimal("0")), Tuple(22, Div(854513, 138), Decimal("6192.12318840579710144927536232")), Tuple(23, 0, Decimal("0")), Tuple(24, Neg(Div(236364091, 2730)), Decimal("-86580.2531135531135531135531136")), Tuple(25, 0, Decimal("0")), Tuple(26, Div(8553103, 6), Decimal("1425517.16666666666666666666667")), Tuple(27, 0, Decimal("0")), Tuple(28, Neg(Div(23749461029, 870)), Decimal("-27298231.0678160919540229885057")), Tuple(29, 0, Decimal("0")), Tuple(30, Div(8615841276005, 14322), Decimal("601580873.900642368384303868175")), Tuple(31, 0, Decimal("0")), Tuple(32, Neg(Div(7709321041217, 510)), Decimal("-15116315767.0921568627450980392")), Tuple(33, 0, Decimal("0")), Tuple(34, Div(2577687858367, 6), Decimal("429614643061.166666666666666667")), Tuple(35, 0, Decimal("0")), Tuple(36, Neg(Div(26315271553053477373, 1919190)), Decimal("-13711655205088.3327721590879486")), Tuple(37, 0, Decimal("0")), Tuple(38, Div(2929993913841559, 6), Decimal("488332318973593.166666666666667")), Tuple(39, 0, Decimal("0")), Tuple(40, Neg(Div(261082718496449122051, 13530)), Decimal("-19296579341940068.1486326681449")), Tuple(41, 0, Decimal("0")), Tuple(42, Div(1520097643918070802691, 1806), Decimal("841693047573682615.000553709856")), Tuple(43, 0, Decimal("0")), Tuple(44, Neg(Div(27833269579301024235023, 690)), Decimal("-40338071854059455413.0768115942")), Tuple(45, 0, Decimal("0")), Tuple(46, Div(596451111593912163277961, 282), Decimal("2115074863808199160560.14539007")), Tuple(47, 0, Decimal("0")), Tuple(48, Neg(Div(5609403368997817686249127547, 46410)), Decimal("-120866265222965259346027.311937")), Tuple(49, 0, Decimal("0")), Tuple(50, Div(495057205241079648212477525, 66), Decimal("7500866746076964366855720.07576")))))

588889

Table of

B_{n}\!\left(x\right)

for

0 \le n \le 10

$n$	$B_{n}\!\left(x\right)$
0	1
1	$x - \frac{1}{2}$
2	${x}^{2} - x + \frac{1}{6}$
3	${x}^{3} - \frac{3}{2} {x}^{2} + \frac{1}{2} x$
4	${x}^{4} - 2 {x}^{3} + {x}^{2} - \frac{1}{30}$
5	${x}^{5} - \frac{5}{2} {x}^{4} + \frac{5}{3} {x}^{3} - \frac{1}{6} x$
6	${x}^{6} - 3 {x}^{5} + \frac{5}{2} {x}^{4} - \frac{1}{2} {x}^{2} + \frac{1}{42}$
7	${x}^{7} - \frac{7}{2} {x}^{6} + \frac{7}{2} {x}^{5} - \frac{7}{6} {x}^{3} + \frac{1}{6} x$
8	${x}^{8} - 4 {x}^{7} + \frac{14}{3} {x}^{6} - \frac{7}{3} {x}^{4} + \frac{2}{3} {x}^{2} - \frac{1}{30}$
9	${x}^{9} - \frac{9}{2} {x}^{8} + 6 {x}^{7} - \frac{21}{5} {x}^{5} + 2 {x}^{3} - \frac{3}{10} x$
10	${x}^{10} - 5 {x}^{9} + \frac{15}{2} {x}^{8} - 7 {x}^{6} + 5 {x}^{4} - \frac{3}{2} {x}^{2} + \frac{5}{66}$

Table data:

\left(n, p\right)

such that

B_{n}\!\left(x\right) = p

Assumptions:

x \in \mathbb{C}

TeX:

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("588889"),
    Description("Table of", BernoulliPolynomial(n, x), "for", LessEqual(0, n, 10)),
    Table(TableRelation(Tuple(n, p), Equal(BernoulliPolynomial(n, x), p)), TableHeadings(n, BernoulliPolynomial(n, x)), TableSplit(1), List(Tuple(0, 1), Tuple(1, Sub(x, Div(1, 2))), Tuple(2, Add(Sub(Pow(x, 2), x), Div(1, 6))), Tuple(3, Add(Sub(Pow(x, 3), Mul(Div(3, 2), Pow(x, 2))), Mul(Div(1, 2), x))), Tuple(4, Sub(Add(Sub(Pow(x, 4), Mul(2, Pow(x, 3))), Pow(x, 2)), Div(1, 30))), Tuple(5, Sub(Add(Sub(Pow(x, 5), Mul(Div(5, 2), Pow(x, 4))), Mul(Div(5, 3), Pow(x, 3))), Mul(Div(1, 6), x))), Tuple(6, Add(Sub(Add(Sub(Pow(x, 6), Mul(3, Pow(x, 5))), Mul(Div(5, 2), Pow(x, 4))), Mul(Div(1, 2), Pow(x, 2))), Div(1, 42))), Tuple(7, Add(Sub(Add(Sub(Pow(x, 7), Mul(Div(7, 2), Pow(x, 6))), Mul(Div(7, 2), Pow(x, 5))), Mul(Div(7, 6), Pow(x, 3))), Mul(Div(1, 6), x))), Tuple(8, Sub(Add(Sub(Add(Sub(Pow(x, 8), Mul(4, Pow(x, 7))), Mul(Div(14, 3), Pow(x, 6))), Mul(Div(7, 3), Pow(x, 4))), Mul(Div(2, 3), Pow(x, 2))), Div(1, 30))), Tuple(9, Sub(Add(Sub(Add(Sub(Pow(x, 9), Mul(Div(9, 2), Pow(x, 8))), Mul(6, Pow(x, 7))), Mul(Div(21, 5), Pow(x, 5))), Mul(2, Pow(x, 3))), Mul(Div(3, 10), x))), Tuple(10, Add(Sub(Add(Sub(Add(Sub(Pow(x, 10), Mul(5, Pow(x, 9))), Mul(Div(15, 2), Pow(x, 8))), Mul(7, Pow(x, 6))), Mul(5, Pow(x, 4))), Mul(Div(3, 2), Pow(x, 2))), Div(5, 66))))),
    Variables(x),
    Assumptions(Element(x, CC)))

522b04

\frac{z}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n} \frac{{z}^{n}}{n !}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0

TeX:

\frac{z}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n} \frac{{z}^{n}}{n !}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("522b04"),
    Formula(Equal(Div(z, Sub(Exp(z), 1)), Sum(Mul(BernoulliB(n), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(z), Mul(2, Pi)), NotEqual(z, 0))))

f79ff0

\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0

TeX:

\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 2 \pi \;\mathbin{\operatorname{and}}\; z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("f79ff0"),
    Formula(Equal(Div(Mul(z, Exp(Mul(x, z))), Sub(Exp(z), 1)), Sum(Mul(BernoulliPolynomial(n, x), Div(Pow(z, n), Factorial(n))), For(n, 0, Infinity)))),
    Variables(z, x),
    Assumptions(And(Element(x, CC), Element(z, CC), Less(Abs(z), Mul(2, Pi)), NotEqual(z, 0))))

555e10

B_{n}\!\left(x\right) = \sum_{k=0}^{n} {n \choose k} B_{n - k} {x}^{k}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

B_{n}\!\left(x\right) = \sum_{k=0}^{n} {n \choose k} B_{n - k} {x}^{k}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("555e10"),
    Formula(Equal(BernoulliPolynomial(n, x), Sum(Mul(Mul(Binomial(n, k), BernoulliB(Sub(n, k))), Pow(x, k)), For(k, 0, n)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

14ecc4

B_{2 n} = {\left(-1\right)}^{n + 1} \frac{2 \left(2 n\right)! \zeta\!\left(2 n\right)}{{\left(2 \pi\right)}^{2 n}}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

B_{2 n} = {\left(-1\right)}^{n + 1} \frac{2 \left(2 n\right)! \zeta\!\left(2 n\right)}{{\left(2 \pi\right)}^{2 n}}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("14ecc4"),
    Formula(Equal(BernoulliB(Mul(2, n)), Mul(Pow(-1, Add(n, 1)), Div(Mul(Mul(2, Factorial(Mul(2, n))), RiemannZeta(Mul(2, n))), Pow(Mul(2, Pi), Mul(2, n)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

4228cd

B_{n}\!\left(z\right) = -n \zeta\!\left(1 - n, z\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

B_{n}\!\left(z\right) = -n \zeta\!\left(1 - n, z\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4228cd"),
    Formula(Equal(BernoulliPolynomial(n, z), Neg(Mul(n, HurwitzZeta(Sub(1, n), z))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC))))

a98234

B_{2 n + 3} = 0

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

B_{2 n + 3} = 0

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("a98234"),
    Formula(Equal(BernoulliB(Add(Mul(2, n), 3)), 0)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

d10873

{\left(-1\right)}^{n} B_{2 n + 2} > 0

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

{\left(-1\right)}^{n} B_{2 n + 2} > 0

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d10873"),
    Formula(Greater(Mul(Pow(-1, n), BernoulliB(Add(Mul(2, n), 2))), 0)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

a1d2d7

B_{n}\!\left(0\right) = B_{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

B_{n}\!\left(0\right) = B_{n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("a1d2d7"),
    Formula(Equal(BernoulliPolynomial(n, 0), BernoulliB(n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

829185

B_{n}\!\left(1\right) = {\left(-1\right)}^{n} B_{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

B_{n}\!\left(1\right) = {\left(-1\right)}^{n} B_{n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("829185"),
    Formula(Equal(BernoulliPolynomial(n, 1), Mul(Pow(-1, n), BernoulliB(n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

03ee0b

B_{n}\!\left(\frac{1}{2}\right) = \left({2}^{1 - n} - 1\right) B_{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

B_{n}\!\left(\frac{1}{2}\right) = \left({2}^{1 - n} - 1\right) B_{n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("03ee0b"),
    Formula(Equal(BernoulliPolynomial(n, Div(1, 2)), Mul(Sub(Pow(2, Sub(1, n)), 1), BernoulliB(n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

8b4f7f

B_{n}\!\left(x + 1\right) = B_{n}\!\left(x\right) + n {x}^{n - 1}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

B_{n}\!\left(x + 1\right) = B_{n}\!\left(x\right) + n {x}^{n - 1}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("8b4f7f"),
    Formula(Equal(BernoulliPolynomial(n, Add(x, 1)), Add(BernoulliPolynomial(n, x), Mul(n, Pow(x, Sub(n, 1)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

c2dcfa

B_{n}\!\left(1 - x\right) = {\left(-1\right)}^{n} B_{n}\!\left(x\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

B_{n}\!\left(1 - x\right) = {\left(-1\right)}^{n} B_{n}\!\left(x\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c2dcfa"),
    Formula(Equal(BernoulliPolynomial(n, Sub(1, x)), Mul(Pow(-1, n), BernoulliPolynomial(n, x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

f80439

B_{n}\!\left(-x\right) = {\left(-1\right)}^{n} \left(B_{n}\!\left(x\right) + n {x}^{n - 1}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

B_{n}\!\left(-x\right) = {\left(-1\right)}^{n} \left(B_{n}\!\left(x\right) + n {x}^{n - 1}\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f80439"),
    Formula(Equal(BernoulliPolynomial(n, Neg(x)), Mul(Pow(-1, n), Add(BernoulliPolynomial(n, x), Mul(n, Pow(x, Sub(n, 1))))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

7adfd6

\int_{a}^{b} B_{n}\!\left(t\right) \, dt = \frac{B_{n + 1}\!\left(b\right) - B_{n + 1}\!\left(a\right)}{n + 1}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

TeX:

\int_{a}^{b} B_{n}\!\left(t\right) \, dt = \frac{B_{n + 1}\!\left(b\right) - B_{n + 1}\!\left(a\right)}{n + 1}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7adfd6"),
    Formula(Equal(Integral(BernoulliPolynomial(n, t), For(t, a, b)), Div(Sub(BernoulliPolynomial(Add(n, 1), b), BernoulliPolynomial(Add(n, 1), a)), Add(n, 1)))),
    Variables(n, a, b),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Element(b, CC))))

05202b

\int_{z}^{z + 1} B_{n}\!\left(t\right) \, dt = {z}^{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\int_{z}^{z + 1} B_{n}\!\left(t\right) \, dt = {z}^{n}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("05202b"),
    Formula(Equal(Integral(BernoulliPolynomial(n, t), For(t, z, Add(z, 1))), Pow(z, n))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

e89eb5

B'_{n}(x) = n B_{n - 1}\!\left(x\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

B'_{n}(x) = n B_{n - 1}\!\left(x\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e89eb5"),
    Formula(Equal(ComplexDerivative(BernoulliPolynomial(n, x), For(x, x, 1)), Mul(n, BernoulliPolynomial(Sub(n, 1), x)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

4aab8a

\sum_{k=1}^{m} {k}^{n} = \frac{B_{n + 1}\!\left(m + 1\right) - B_{m + 1}}{m + 1}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

TeX:

\sum_{k=1}^{m} {k}^{n} = \frac{B_{n + 1}\!\left(m + 1\right) - B_{m + 1}}{m + 1}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4aab8a"),
    Formula(Equal(Sum(Pow(k, n), For(k, 1, m)), Div(Sub(BernoulliPolynomial(Add(n, 1), Add(m, 1)), BernoulliB(Add(m, 1))), Add(m, 1)))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

ff190c

\left(B_{2 n} + \sum_{\left(p - 1\right) \mid 2 n} \frac{1}{p}\right) \in \mathbb{Z}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\left(B_{2 n} + \sum_{\left(p - 1\right) \mid 2 n} \frac{1}{p}\right) \in \mathbb{Z}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number
`PrimeSum`	$\sum_{p} f(p)$	Sum over primes
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ff190c"),
    Formula(Element(Parentheses(Add(BernoulliB(Mul(2, n)), PrimeSum(Div(1, p), For(p), Divides(Parentheses(Sub(p, 1)), Mul(2, n))))), ZZ)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

c33e2b

\left(B_{2 n} \prod_{\left(p - 1\right) \mid 2 n} \frac{1}{p}\right) \in \mathbb{Z}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\left(B_{2 n} \prod_{\left(p - 1\right) \mid 2 n} \frac{1}{p}\right) \in \mathbb{Z}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number
`PrimeProduct`	$\prod_{p} f(p)$	Product over primes
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("c33e2b"),
    Formula(Element(Parentheses(Mul(BernoulliB(Mul(2, n)), PrimeProduct(Div(1, p), For(p), Divides(Parentheses(Sub(p, 1)), Mul(2, n))))), ZZ)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

0f02a5

\left|B_{2 n}\right| < \left(1 + \frac{1}{n}\right) \frac{2 \left(2 n\right)!}{{\left(2 \pi\right)}^{2 n}}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\left|B_{2 n}\right| < \left(1 + \frac{1}{n}\right) \frac{2 \left(2 n\right)!}{{\left(2 \pi\right)}^{2 n}}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("0f02a5"),
    Formula(Less(Abs(BernoulliB(Mul(2, n))), Mul(Add(1, Div(1, n)), Div(Mul(2, Factorial(Mul(2, n))), Pow(Mul(2, Pi), Mul(2, n)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

3a1316

\left|B_{2 n}\right| < \left(1 + \frac{1}{n}\right) 4 \sqrt{\pi n} {\left(\frac{n}{\pi e}\right)}^{2 n}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\left|B_{2 n}\right| < \left(1 + \frac{1}{n}\right) 4 \sqrt{\pi n} {\left(\frac{n}{\pi e}\right)}^{2 n}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BernoulliB`	$B_{n}$	Bernoulli number
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`ConstE`	$e$	The constant e (2.718...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("3a1316"),
    Formula(Less(Abs(BernoulliB(Mul(2, n))), Mul(Mul(Mul(Add(1, Div(1, n)), 4), Sqrt(Mul(Pi, n))), Pow(Div(n, Mul(Pi, ConstE)), Mul(2, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

1d2f4a

\left|B_{2 n}\right| > \frac{2 \left(2 n\right)!}{{\left(2 \pi\right)}^{2 n}}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\left|B_{2 n}\right| > \frac{2 \left(2 n\right)!}{{\left(2 \pi\right)}^{2 n}}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("1d2f4a"),
    Formula(Greater(Abs(BernoulliB(Mul(2, n))), Div(Mul(2, Factorial(Mul(2, n))), Pow(Mul(2, Pi), Mul(2, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

69ca86

\left|B_{2 n}\right| > 4 \sqrt{\pi n} {\left(\frac{n}{\pi e}\right)}^{2 n}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\left|B_{2 n}\right| > 4 \sqrt{\pi n} {\left(\frac{n}{\pi e}\right)}^{2 n}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BernoulliB`	$B_{n}$	Bernoulli number
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`ConstE`	$e$	The constant e (2.718...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("69ca86"),
    Formula(Greater(Abs(BernoulliB(Mul(2, n))), Mul(Mul(4, Sqrt(Mul(Pi, n))), Pow(Div(n, Mul(Pi, ConstE)), Mul(2, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

292d70

\left|B_{2 n}\!\left(x\right)\right| \le \left|B_{2 n}\right|

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \left[0, 1\right]

TeX:

\left|B_{2 n}\!\left(x\right)\right| \le \left|B_{2 n}\right|

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \left[0, 1\right]

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("292d70"),
    Formula(LessEqual(Abs(BernoulliPolynomial(Mul(2, n), x)), Abs(BernoulliB(Mul(2, n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, ClosedInterval(0, 1)))))

4246ae

\left|B_{2 n}\!\left(x\right)\right| < \left|B_{2 n}\right|

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left(0, 1\right)

TeX:

\left|B_{2 n}\!\left(x\right)\right| < \left|B_{2 n}\right|

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left(0, 1\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`BernoulliB`	$B_{n}$	Bernoulli number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("4246ae"),
    Formula(Less(Abs(BernoulliPolynomial(Mul(2, n), x)), Abs(BernoulliB(Mul(2, n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, OpenInterval(0, 1)))))

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