aarjaneiro/planetmath huggingface dataset:
Farey sequence

Farey sequence

The

n

’th

Farey sequence

is the ascending

sequence

of all

rationals

{

0

≤

a

b

≤

1

:

b

≤

n

}

.

The first 5 Farey sequences are

1

0

1

<

1

1

2

0

1

<

1

2

<

1

1

3

0

1

<

1

3

<

1

2

<

2

3

<

1

1

4

0

1

<

1

4

<

1

3

<

1

2

<

2

3

<

3

4

<

1

1

5

0

1

<

1

5

<

1

4

<

1

3

<

2

5

<

1

2

<

3

5

<

2

3

<

3

4

<

4

5

<

1

1

Farey sequences are a singularly useful tool in understanding the

convergents

that appear in

continued fractions

. The convergents for any

irrational

α

can be found: they are precisely the closest number to

α

on the sequences

F

n

.

It is also of value to look at the sequences

F

n

as

n

grows. If

a

b

and

c

d

are reduced

representations

of

adjacent

terms in some Farey sequence

F

n

(where

b

,

d

≤

n

), then they are

adjacent fractions

; their

difference

is the least possible:

|

a

b

-

c

d

|

=

1

b

⁢

d

.

Furthermore, the

first

fraction

to appear between the two in a Farey sequence is

a

+

c

b

+

d

, in sequence

F

b

+

d

, and (as written here) this fraction is already reduced.

An alternate view of the “dynamics” of how Farey sequences develop is given by

Stern-Brocot trees

.

Title

Farey sequence

Canonical

name

FareySequence

Date of creation

2013-03-22 12:47:16

Last modified on

2013-03-22 12:47:16

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

7

Author

ariels (338)

Entry type

Definition

Classification

msc 11B57

Classification

msc 11A55

Related topic

ContinuedFraction

Generated on Fri Feb 9 15:28:19 2018 by

LaTeXML
an algebraic identity leading to Wilson’s theorem

an algebraic identity leading to Wilson’s theorem

For any

positive integer

n

and any real or

complex number

x

,

∑

k

=

0

n

(

-

1

)

k

⁢

(

n

k

)

⁢

(

x

-

k

)

n

=

n

!

Furthermore, if

n

>

m

then

∑

k

=

0

n

(

-

1

)

k

⁢

(

n

k

)

⁢

(

x

-

k

)

m

=

0

References

1

S. M Ruiz.

An algebraic identity leading to Wilson’s theorem.

The Mathemtical Gazette

, 80(489):579–582, November 1996.

http://arxiv.org/abs/math.GM/0406086

math.GM/0406086.

Title

an algebraic identity leading to Wilson’s theorem

Canonical name

AnAlgebraicIdentityLeadingToWilsonsTheorem

Date of creation

2013-03-22 14:31:38

Last modified on

2013-03-22 14:31:38

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

16

Author

GeraW (6138)

Entry type

Result

Classification

msc 11B65

Classification

msc 05A10

Related topic

Factorial

Related topic

WilsonsTheorem

Generated on Fri Feb 9 15:28:21 2018 by

LaTeXML
binomial theorem

binomial theorem

The

binomial theorem

is a

formula

for the expansion of

(

a

+

b

)

n

, for

n

a

positive integer

and

a

and

b

any two real (or complex) numbers, into a sum of powers of

a

and

b

. More precisely,

(

a

+

b

)

n

=

a

n

+

(

n

1

)

⁢

a

n

-

1

⁢

b

+

(

n

2

)

⁢

a

n

-

2

⁢

b

2

+

⋯

+

b

n

.

For example, if

n

is 3 or 4, we have:

(

a

+

b

)

3

=

a

3

+

3

⁢

a

2

⁢

b

+

3

⁢

a

⁢

b

2

+

b

3

(

a

+

b

)

4

=

a

4

+

4

⁢

a

3

⁢

b

+

6

⁢

a

2

⁢

b

2

+

4

⁢

a

⁢

b

3

+

b

4

.

This result actually holds more generally if

a

and

b

belong to a

commutative

rig (

http://planetmath.org/Rig

).

Title

binomial theorem

Canonical name

BinomialTheorem

Date of creation

2013-03-22 11:46:47

Last modified on

2013-03-22 11:46:47

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

17

Author

KimJ (5)

Entry type

Theorem

Classification

msc 11B65

Classification

msc 06F25

Classification

msc 03E20

Related topic

BinomialFormula

Related topic

BinomialCoefficient

Related topic

BernoulliDistribution2

Related topic

UsingThePrimitiveElementOfBiquadraticField

Generated on Fri Feb 9 15:28:32 2018 by

LaTeXML
binomial theorem, proof of

binomial theorem, proof of

Proposition

.

Let a and b be commuting elements of some rig. Then

(

a

+

b

)

n

=

∑

k

=

0

n

(

n

k

)

⁢

a

k

⁢

b

n

-

k

,

where the

(

n

k

)

are

binomial coefficients

.

Proof.

Each term in the expansion of

(

a

+

b

)

n

is obtained by making

n

decisions of whether to use

a

or

b

as a factor. Moreover,
any

sequence

of

n

such decisions yields a term in the expansion.
So the expandsion of

(

a

+

b

)

n

is precisely the sum of all the

ab

-words of length

n

, where each word appears exactly once.

Since

a

and

b

commute, we can reduce each term via

rewrite
rules

of the form

b

n

⁢

a

↦

a

⁢

b

n

to a term in which the

a

factors precede all the

b

factors. This produces a term of the
form

a

k

⁢

b

n

-

k

for some

k

, where we use the expressions

a

0

⁢

b

n

and

a

n

⁢

b

0

to denote

b

n

and

a

n

respectively. For
example, reducing the word

b

⁢

a

⁢

b

⁢

a

⁢

b

2

⁢

a

⁢

b

⁢

a

yields

a

4

⁢

b

5

, via the
following

reduction

.

b

⁢

a

⁢

b

⁢

a

⁢

b

2

⁢

a

⁢

b

⁢

a

↦

a

⁢

b

⁢

a

⁢

b

⁢

a

⁢

b

2

⁢

a

⁢

b

↦

a

2

⁢

b

⁢

a

⁢

b

⁢

a

⁢

b

3

↦

a

3

⁢

b

⁢

a

⁢

b

4

↦

a

4

⁢

b

5

.

After performing this rewriting process, we collect like terms. Let
us illustrate this with the case

n

= 3.

(

a

+

b

)

3

=

a

⁢

a

⁢

a

+

a

⁢

a

⁢

b

+

a

⁢

b

⁢

a

+

a

⁢

b

⁢

b

+

b

⁢

a

⁢

a

+

b

⁢

a

⁢

b

+

b

⁢

b

⁢

a

+

b

⁢

b

⁢

b

=

a

3

+

a

2

⁢

b

+

a

2

⁢

b

+

a

⁢

b

2

+

a

2

⁢

b

+

a

⁢

b

2

+

a

⁢

b

2

+

b

3

=

a

3

+

3

⁢

a

2

⁢

b

+

3

⁢

a

⁢

b

2

+

b

3

.

To determine the coefficient of a

reduced

term, it suffices to
determine how many

ab

-words have that reduction. Since reducing
a term only changes the positions of

a

s and

b

s and not their
number, all the

ab

-words where

k

of the letters are

b

s
and

n-k

are

a

s, for

0

≤

k

≤

n

, have the same
normalization. But there are exactly

(

n

k

)

such

ab

-words, since there are

(

n

k

)

ways to select

k

positions out of

n

to place

a

s in an

ab

-word of length

n

. This shows that the coefficient of the

a

n

term is

(

n

0

)

=

1

, the coefficient of the

b

n

term is

(

n

n

)

=

1

, and that the coefficient of the

a

k

⁢

b

n

-

k

term
is

(

n

k

)

.
∎

Title

binomial theorem

, proof of

Canonical name

BinomialTheoremProofOf

Date of creation

2013-03-22 15:03:54

Last modified on

2013-03-22 15:03:54

Owner

mps (409)

Last modified by

mps (409)

Numerical id

15

Author

mps (409)

Entry type

Proof

Classification

msc 11B65

Generated on Fri Feb 9 15:28:35 2018 by

LaTeXML
generalized binomial coefficients

generalized binomial coefficients

The

binomial coefficients

(

n

r

)

=

n

!

(

n

-

r

)

!

⁢

r

!

,

(1)

where

n

is a non-negative

integer

and

r

∈

{

0

,

1

,

2

,

…

,

n

}

,
can be generalized for all integer and non-integer values of

n

by using the

reduced

(

http://planetmath.org/Division

) form

(

n

r

)

=

n

⁢

(

n

-

1

)

⁢

(

n

-

2

)

⁢

…

⁢

(

n

-

r

+

1

)

r

!

;

(2)

here

r

may be any non-negative integer.  Then Newton’s

binomial series

(

http://planetmath.org/BinomialFormula

) gets the form

(

1

+

z

)

α

=

∑

r

=

0

∞

(

α

r

)

⁢

z

r

=

1

+

(

α

1

)

⁢

z

+

(

α

2

)

⁢

z

2

+

⋯

(3)

It is not hard to show that the

radius of convergence

of this series is 1.  This

series expansion

is valid for every

complex number

α

when

|

z

|

<

1

,  and it presents such a branch (

http://planetmath.org/GeneralPower

) of the power (

http://planetmath.org/GeneralPower

)

(

1

+

z

)

α

which gets the value 1 in the point

z

=

0

.

In the case that

α

is a non-negative integer and

r

is great enough, one factor in the

numerator

of

(

α

r

)

=

α

⁢

(

α

-

1

)

⁢

(

α

-

2

)

⁢

…

⁢

(

α

-

r

+

1

)

r

!

(4)

vanishes, and hence the corresponding binomial coefficient

(

α

r

)

equals to zero; accordingly also all following binomial coefficients with a greater

r

are equal to zero.  It means that the series is left to being a finite sum, which gives the

binomial theorem

.

For all

complex

values of

α

,

β

and non-negative integer values of

r

,

s

, the

Pascal’s

formula

(

α

r

)

+

(

α

r

+

1

)

=

(

α

+

1

r

+

1

)

(5)

and

Vandermonde’s convolution

∑

r

=

0

s

(

α

r

)

⁢

(

β

s

-

r

)

=

(

α

+

β

s

)

(6)

hold (the latter is proved by expanding the power

(

1

+

z

)

α

+

β

to series).  Cf. Pascal’s rule and

Vandermonde identity

.

Title

generalized binomial coefficients

Canonical name

GeneralizedBinomialCoefficients

Date of creation

2013-03-22 14:41:53

Last modified on

2013-03-22 14:41:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

26

Author

pahio (2872)

Entry type

Definition

Classification

msc 11B65

Classification

msc 05A10

Related topic

BinomialFormula

Related topic

GeneralPower

Related topic

BinomialFormulaForNegativeIntegerPowers

Defines

Pascal’s formula

Defines

Vandermonde’s formula

Generated on Fri Feb 9 15:28:38 2018 by

LaTeXML
generalized Ruiz’s identity

generalized Ruiz’s identity

For

i

,

j

=

1

,

2

,

…

, consider the

polynomials

c

i

,

j

⁢

(

x

)

=

(

x

+

i

)

j

-

(

x

+

i

-

1

)

j

(

x

is the

indeterminate

). Then, for every positive natural number

n

,

det

⁡

(

c

i

,

j

)

i

,

j

=

1

n

=

∏

k

=

1

n

k

!

Title

generalized Ruiz’s

identity

Canonical

name

GeneralizedRuizsIdentity

Date of creation

2013-03-22 14:31:45

Last modified on

2013-03-22 14:31:45

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

7

Author

GeraW (6138)

Entry type

Derivation

Classification

msc 11B65

Classification

msc 05A10

Related topic

determinant2

Related topic

Determinant2

Generated on Fri Feb 9 15:28:40 2018 by

LaTeXML
Lucas’s theorem

Lucas’s theorem

Let

m

,

n

∈

ℕ

-

{

0

}

be two

natural numbers

. If

p

is a

prime number

and :

m

=

a

k

⁢

p

k

+

a

k

-

1

⁢

p

k

-

1

+

⋯

+

a

1

⁢

p

+

a

0

,

n

=

b

k

⁢

p

k

+

b

k

-

1

⁢

p

k

-

1

+

⋯

+

b

1

⁢

p

+

b

0

are the base-p expansions of

m

and

n

, then the following

congruence

is true :

(

m

n

)

≡

(

a

0

b

0

)

⁢

(

a

1

b

1

)

⁢

⋯

⁢

(

a

k

b

k

)

⁢

(

𝚖𝚘𝚍𝚙

)

Note : the

binomial coefficient

is defined in the usual way , namely :

(

x

y

)

=

x

!

y

!

⁢

(

x

-

y

)

!

if

x

≥

y

and

0

otherwise (of course , x and y are natural numbers).

Title

Lucas’s

theorem

Canonical name

LucassTheorem

Date of creation

2013-03-22 13:17:31

Last modified on

2013-03-22 13:17:31

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11B65

Generated on Fri Feb 9 15:28:42 2018 by

LaTeXML
proof of generalized Ruiz’s identity

proof of generalized Ruiz’s identity

Theorem

.

Consider the

polynomials

c

i

,

j

⁢

(

x

)

=

(

x

+

i

)

j

-

(

x

+

i

-

1

)

j

. Then, for every positive natural number

n

,

det

⁡

(

c

i

,

j

)

i

,

j

=

1

n

=

∏

k

=

1

n

k

!

Proof.

Consider the matrices

M

,

C

defined by

M

i

,

j

=

(

-

1

)

j

⁢

(

i

-

1

j

-

1

)

and

C

i

,

j

=

(

x

+

i

)

j

-

(

x

+

i

-

1

)

j

.

(

M

⁢

C

)

i

,

j

=

∑

k

=

1

n

(

-

1

)

k

⁢

(

i

-

1

k

-

1

)

⁢

(

(

x

+

k

)

j

-

(

x

+

k

-

1

)

j

)

=

∑

k

=

0

n

(

-

1

)

k

⁢

(

(

i

-

1

k

-

1

)

+

(

i

-

1

k

)

)

⁢

(

x

+

k

)

j

=

∑

k

=

0

n

(

-

1

)

k

⁢

(

i

k

)

⁢

(

x

+

k

)

j

=

(

-

1

)

j

⁢

∑

k

=

0

i

(

-

1

)

k

⁢

(

i

k

)

⁢

(

-

x

-

k

)

j

Therefore, by Ruiz’s

identity

,

(

M

⁢

C

)

i

,

i

=

(

-

1

)

i

⁢

i

!

for every

i

∈

{

1

,

…

,

d

}

and

(

M

⁢

C

)

i

,

j

=

0

for every

i

,

j

∈

{

1

,

…

,

n

}

such that

i

>

j

. This
means that

M

⁢

C

is an

upper triangular matrix

whose main diagonal is

-

1

!

,

2

!

,

-

3

!

,

…

,

(

-

1

)

n

⁢

n

!

. Since the

determinant

of such a matrix is
the

product

of the elements in the main diagonal, we get that

det

⁡

M

⁢

C

=

(

-

1

)

n

⁢

∏

k

=

1

n

k

!

. It is easy to see that

M

itself is lower
triangular with determinant

(

-

1

)

n

. Therefore

det

⁡

C

=

∏

k

=

1

n

k

!

.
∎

Title

proof of generalized Ruiz’s identity

Canonical name

ProofOfGeneralizedRuizsIdentity

Date of creation

2013-03-22 14:32:02

Last modified on

2013-03-22 14:32:02

Owner

GeraW (6138)

Last modified by

GeraW (6138)

Numerical id

8

Author

GeraW (6138)

Entry type

Proof

Classification

msc 11B65

Classification

msc 05A10

Generated on Fri Feb 9 15:28:45 2018 by

LaTeXML
proof of Lucas’s theorem by binomial expansion

proof of Lucas’s theorem by binomial expansion

We work with polynomials in

x

over the integers modulo

p

.

By
the

binomial theorem

we have

(

1

+

x

)

p

=

1

+

x

p

. More
generally, by

induction

on

i

we have

(

1

+

x

)

p

i

=

1

+

x

p

i

.

Hence the following holds:

(

1

+

x

)

n

=

(

1

+

x

)

[

∑

i

=

0

k

a

i

⁢

p

i

]

=

∏

i

=

0

k

(

1

+

x

p

i

)

a

i

=

∏

i

=

0

k

∑

b

=

0

a

i

(

a

i

b

)

⁢

x

b

⁢

p

i

Then the coefficient on

x

m

on the

left hand side

is

(

n

m

)

.

As

m

is uniquely base

p

, the coefficient on

x

m

on the right hand side is

∏

i

=

0

k

(

a

i

b

i

)

.

Equating the coefficients on

x

m

on either therefore yields
the result.

Title

proof of Lucas’s

theorem

by binomial expansion

Canonical name

ProofOfLucassTheoremByBinomialExpansion

Date of creation

2013-03-22 18:19:59

Last modified on

2013-03-22 18:19:59

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

4

Author

whm22 (2009)

Entry type

Proof

Classification

msc 11B65

Generated on Fri Feb 9 15:28:47 2018 by

LaTeXML
(n/r) is an integer

(

n

r

)

is an integer

Theorem 1

.

For

n

≥

r

≥

0

, the

binomial coefficient

(

n

r

)

is an

integer

.

Proof.

The proof is by

induction

on

n

. For

n

=

0

, the
claim is clear. Thus, suppose the claim holds for

n

≥

1

.
For

r

=

1

,

…

,

n

, Pascal’s rule gives

(

n

+

1

r

)

=

(

n

r

)

+

(

n

r

-

1

)

.

That is,

(

n

+

1

1

)

,

…

,

(

n

+

1

n

)

are
integers.
Since

(

n

+

1

0

)

=

1

,

(

n

+

1

n

+

1

)

=

1

the proof is

complete

.
∎

Title

(

n

r

)

is an integer

Canonical name

nchooseRIsAnInteger

Date of creation

2013-03-22 15:02:01

Last modified on

2013-03-22 15:02:01

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 11B65

Classification

msc 05A10

Generated on Fri Feb 9 15:28:51 2018 by

LaTeXML
alternate form of sum of rth powers of the first n positive integers

alternate form of sum of

r

th powers of the first

n

positive integers

We will show that

∑

k

=

0

n

k

r

=

∫

1

n

+

1

b

r

⁢

(

x

)

⁢

𝑑

x

We need two basic facts. First, a

property

of the

Bernoulli polynomials

is that

b

r

′

⁢

(

x

)

=

r

⁢

b

r

-

1

⁢

(

x

)

. Second, the Bernoulli polynomials can be written as

b

r

⁢

(

x

)

=

∑

k

=

1

r

(

r

k

)

⁢

B

r

-

k

⁢

x

k

+

B

r

We then have

∫

1

n

+

1

b

r

⁢

(

x

)

=

1

r

+

1

⁢

(

b

r

+

1

⁢

(

n

+

1

)

-

b

r

+

1

⁢

(

1

)

)

=

1

r

+

1

⁢

∑

k

=

0

r

+

1

(

r

+

1

k

)

⁢

B

r

+

1

-

k

⁢

(

(

n

+

1

)

k

-

1

)

=

1

r

+

1

⁢

∑

k

=

1

r

+

1

(

r

+

1

k

)

⁢

B

r

+

1

-

k

⁢

(

n

+

1

)

k

Now reverse the order of

summation

(i.e. replace

k

by

r

+

1

-

k

) to get

∫

1

n

+

1

b

r

⁢

(

x

)

=

1

r

+

1

⁢

∑

k

=

0

r

(

r

+

1

r

+

1

-

k

)

⁢

B

k

⁢

(

n

+

1

)

r

+

1

-

k

=

1

r

+

1

⁢

∑

k

=

0

r

(

r

+

1

k

)

⁢

B

r

⁢

(

n

+

1

)

r

+

1

-

k

which is equal to

∑

k

=

0

n

k

r

(see the parent (

http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers

) article).

Title

alternate form of sum of

r

th powers of the first

n

positive integers

Canonical

name

AlternateFormOfSumOfRthPowersOfTheFirstNPositiveIntegers

Date of creation

2013-03-22 17:46:10

Last modified on

2013-03-22 17:46:10

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Proof

Classification

msc 11B68

Classification

msc 05A15

Generated on Fri Feb 9 15:28:54 2018 by

LaTeXML
Bernoulli number

Bernoulli number

Let

B

r

be the

r

th

Bernoulli polynomial

. Then the

r

th

Bernoulli number

is

B

r

:=

B

r

⁢

(

0

)

.

This means, in particular, that the Bernoulli numbers are given by an

exponential generating function

in the following way:

∑

r

=

0

∞

B

r

⁢

y

r

r

!

=

y

e

y

-

1

and, in fact, the Bernoulli numbers are usually defined as the

coefficients

that appear in such expansion.

Observe that this

generating function

can be rewritten:

y

e

y

-

1

=

y

2

⁢

e

y

+

1

e

y

-

1

-

y

2

=

(

y

/

2

)

⁢

(

tanh

⁡

(

y

/

2

)

-

1

)

.

Since

tanh

is an

odd function

, one can see that

B

2

⁢

r

+

1

=

0

for

r

≥

1

. Numerically,

B

0

=

1

,

B

1

=

-

1

2

,

B

2

=

1

6

,

B

4

=

-

1

30

,

⋯

These combinatorial numbers occur in a number of contexts; the most elementary is perhaps that they occur in the formulas for the sum of the

r

th powers of the first

n

positive integers

(

http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers

). They also occur in the Maclaurin expansion for the

tangent function

and in the

Euler-Maclaurin summation formula

.

Title

Bernoulli number

Canonical

name

BernoulliNumber

Date of creation

2013-03-22 11:45:58

Last modified on

2013-03-22 11:45:58

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

14

Author

alozano (2414)

Entry type

Definition

Classification

msc 11B68

Classification

msc 49J24

Classification

msc 49J22

Classification

msc 49J20

Classification

msc 49J15

Related topic

GeneralizedBernoulliNumber

Related topic

BernoulliPolynomials

Related topic

SumOfKthPowersOfTheFirstNPositiveIntegers

Related topic

EulerMaclaurinSummationFormula

Related topic

ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers

Related topic

TaylorSeriesViaDivision

Related topic

BernoulliPolynomialsAndNumbers

Related topic

EulerNumbers2

Generated on Fri Feb 9 15:28:56 2018 by

LaTeXML
Bernoulli periodic function

Bernoulli periodic function

Let

b

r

be the

r

⁢

th

Bernoulli polynomial

. Then the

r

⁢

th

Bernoulli periodic function

B

r

⁢

(

x

)

is defined as the

periodic function

of period 1 which coincides with

b

r

on

[

0

,

1

]

.

Title

Bernoulli periodic function

Canonical name

BernoulliPeriodicFunction

Date of creation

2013-03-22 11:45:54

Last modified on

2013-03-22 11:45:54

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

10

Author

KimJ (5)

Entry type

Definition

Classification

msc 11B68

Generated on Fri Feb 9 15:28:57 2018 by

LaTeXML
Bernoulli polynomial

Bernoulli polynomial

The

Bernoulli polynomials

are the

sequence

{

b

r

⁢

(

x

)

}

r

=

0

∞

of

polynomials

defined on

[

0

,

1

]

by the conditions:

b

0

⁢

(

x

)

=

1

,

b

r

′

⁢

(

x

)

=

r

⁢

b

r

-

1

⁢

(

x

)

,

r

≥

1

,

∫

0

1

b

r

⁢

(

x

)

⁢

𝑑

x

=

0

,

r

≥

1

These

assumptions

imply the

identity

∑

r

=

0

∞

b

r

⁢

(

x

)

⁢

y

r

r

!

=

y

⁢

e

x

⁢

y

e

y

-

1

allowing us to calculate the

b

r

. We have

b

0

⁢

(

x

)

=

1

b

1

⁢

(

x

)

=

x

-

1

2

b

2

⁢

(

x

)

=

x

2

-

x

+

1

6

b

3

⁢

(

x

)

=

x

3

-

3

2

⁢

x

2

+

1

2

⁢

x

b

4

⁢

(

x

)

=

x

4

-

2

⁢

x

3

+

x

2

-

1

30

⋮

Title

Bernoulli polynomial

Canonical name

BernoulliPolynomial

Date of creation

2013-03-22 11:45:51

Last modified on

2013-03-22 11:45:51

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

12

Author

KimJ (5)

Entry type

Definition

Classification

msc 11B68

Classification

msc 65-01

Related topic

BernoulliNumber

Generated on Fri Feb 9 15:28:59 2018 by

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Bernoulli polynomials and numbers

Bernoulli polynomials and numbers

For

n

=

0

,

1

,

2

,

…

,  the

Bernoulli polynomial

may be defined as the uniquely determined

polynomial

b

n

⁢

(

x

)

satisfying

∫

x

x

+

1

b

n

⁢

(

t

)

⁢

𝑑

t

=

x

n

.

(1)

The

constant term

of

b

n

⁢

(

x

)

is the

n

th

Bernoulli number

B

n

.

The Bernoulli polynomial is often denoted also

B

n

⁢

(

x

)

.

The uniqueness of the solution

b

n

⁢

(

x

)

in (1) is justificated by the

Lemma.

For any polynomial

f

⁢

(

x

)

, there exists a unique polynomial

g

⁢

(

x

)

with the same degree satisfying

∫

x

x

+

1

g

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

x

)

.

(2)

Proof.

For every

n

=

0

,

1

,

2

,

…

,  the polynomial

g

n

(

x

)

=

:

∫

x

x

+

1

t

n

d

t

=

(

x

+

1

)

n

+

1

-

x

n

+

1

n

+

1

is monic and its degree is

n

.  If the coefficient of

x

n

in

f

⁢

(

x

)

is

a

0

, then the

difference

f

⁢

(

x

)

-

a

0

⁢

g

n

⁢

(

x

)

is a polynomial of degree

n

-

1

.  Correspondingly we obtain

f

⁢

(

x

)

-

a

0

⁢

g

n

⁢

(

x

)

-

a

1

⁢

g

n

-

1

⁢

(

x

)

having the degree

n

-

2

and so on.  Finally we see that

f

⁢

(

x

)

-

a

0

⁢

g

n

⁢

(

x

)

-

a

1

⁢

g

n

-

1

⁢

(

x

)

-

…

-

a

n

⁢

g

0

⁢

(

x

)

must be the

zero polynomial

.  Therefore

f

⁢

(

x

)

=

a

0

⁢

g

n

⁢

(

x

)

+

a

1

⁢

g

n

-

1

⁢

(

x

)

+

…

+

a

n

⁢

g

0

⁢

(

x

)

=

∑

i

=

0

n

a

i

⁢

g

n

-

i

⁢

(

x

)

=

∑

i

=

0

n

a

i

⁢

∫

x

x

+

1

t

n

-

i

⁢

𝑑

t

=

∫

x

x

+

1

∑

i

=

0

n

a

i

⁢

t

n

-

i

⁢

d

⁢

t

whence we have

g

⁢

(

x

)

=

∑

i

=

0

n

a

i

⁢

x

n

-

i

.

The proof implies also that the coefficients of

g

⁢

(

x

)

are

rational

, if the coefficients of

f

⁢

(

x

)

are such.  So we know that all Bernoulli polynomials have only rational coefficients.

The relation (1) implies easily, that the Bernoulli polynomials form an Appell sequence.

References

1

М. М. Постников:

Введение  в  теорию  алгебраических
чисел

.  Издательство
‘‘Наука’’. Москва (1982).

English

translation

:

M. M. Postnikov:

Introduction to

algebraic number theory

. Science Publs (‘‘Nauka’’).
Moscow (1982).

Title

Bernoulli polynomials and numbers

Canonical

name

BernoulliPolynomialsAndNumbers

Date of creation

2013-03-22 17:58:43

Last modified on

2013-03-22 17:58:43

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 11B68

Synonym

Bernoulli numbers and polynomials

Related topic

BernoulliNumber

Related topic

CoefficientsOfBernoulliPolynomials

Related topic

TaylorSeriesViaDivision

Related topic

ReferenceRelatedToBernoulliPolynomialsAndNumbers

Related topic

EulerPolynomial

Defines

Bernoulli polynomial

Defines

Bernoulli number

Generated on Fri Feb 9 15:29:06 2018 by

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