aarjaneiro/planetmath huggingface dataset:
coefficients of Bernoulli polynomials

coefficients of Bernoulli polynomials

The

coefficient

of

x

k

in

b

r

⁢

(

x

)

for

k

=

1

,

2

,

…

,

r

is

(

r

k

)

⁢

B

r

-

k

.

The proof is by

induction

on

r

. For

r

=

1

, note that

b

1

⁢

(

x

)

=

x

-

1

2

, so that

[

x

]

⁢

b

1

⁢

(

x

)

=

1

=

(

1

1

)

⁢

B

0

.

Writing

[

x

k

]

⁢

f

⁢

(

x

)

for the coefficient of

x

k

in a

polynomial

f

⁢

(

x

)

, note that for

k

=

1

,

2

,

…

,

r

,

[

x

k

]

⁢

b

r

⁢

(

x

)

=

1

k

⁢

[

x

k

-

1

]

⁢

b

r

′

⁢

(

x

)

=

r

k

⁢

[

x

k

-

1

]

⁢

b

r

-

1

⁢

(

x

)

since

b

r

′

⁢

(

x

)

=

r

⁢

b

r

-

1

⁢

(

x

)

. By induction,

r

k

⁢

[

x

k

-

1

]

⁢

b

r

-

1

⁢

(

x

)

=

r

k

⁢

(

r

-

1

k

-

1

)

⁢

B

r

-

k

=

(

r

k

)

⁢

B

r

-

k

Thus the

Bernoulli polynomials

can be written

b

r

⁢

(

x

)

=

∑

k

=

1

r

(

r

k

)

⁢

B

r

-

k

⁢

x

k

+

B

r

Title

coefficients of Bernoulli polynomials

Canonical name

CoefficientsOfBernoulliPolynomials

Date of creation

2013-03-22 17:46:08

Last modified on

2013-03-22 17:46:08

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Derivation

Classification

msc 11B68

Related topic

BernoulliPolynomialsAndNumbers

Generated on Fri Feb 9 15:29:08 2018 by

LaTeXML
congruence of Clausen and von Staudt

congruence of Clausen and von Staudt

Let

B

k

denote the

k

th

Bernoulli number

:

B

0

=

1

,

B

1

=

-

1

2

,

B

2

=

1

6

,

B

3

=

0

,

B

4

=

-

1

30

,

…

In fact,

B

k

=

0

for all odd

k

≥

3

, so we will only consider

B

k

for even

k

. The following is a well-known

congruence

, due to Thomas Clausen and Karl von Staudt.

Theorem

(Congruence of Clausen and von Staudt).

For an

even integer

k

≥

2

,

B

k

≡

-

∑

p

prime

,

(

p

-

1

)

|

k

1

p

mod

ℤ

where the sum is over all primes

p

such that

(

p

-

1

)

divides

k

. In other words, there exists an

integer

n

k

such that

B

k

=

n

k

-

∑

p

prime

,

(

p

-

1

)

|

k

1

p

.

For example:

B

2

=

1

6

=

1

-

1

2

-

1

3

,

B

4

=

-

1

30

=

1

-

1

2

-

1

3

-

1

5

.

Sometimes the

theorem

is stated in this alternative form:

Corollary

.

For an even integer

k

≥

2

and any prime

p

the

product

p

⁢

B

k

is

p

-

integral

, that is,

p

⁢

B

k

is a

rational number

t

/

s

(in

lowest terms

) such that

p

does not divide

s

. Moreover:

p

⁢

B

k

≡

{

-

1

mod

p

,

if

(

p

-

1

)

divides

k

;

0

mod

p

,

if

(

p

-

1

)

does not divide

k

.

Title

congruence of Clausen and von Staudt

Canonical name

CongruenceOfClausenAndVonStaudt

Date of creation

2013-03-22 15:11:58

Last modified on

2013-03-22 15:11:58

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11B68

Synonym

Staudt-Clausen theorem

Synonym

von Staudt-Clausen theorem

Related topic

KummersCongruence

Related topic

OddBernoulliNumbersAreZero

Generated on Fri Feb 9 15:29:11 2018 by

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Euler numbers

Euler numbers

Euler numbers

E

n

have the

generating function

1

cosh

⁡

x

such that

1

cosh

⁡

x

=

:

∑

n

=

0

∞

E

n

n

!

x

n

.

They are

integers

but have no expression for calculating them.  Their only are that the numbers with odd index (

http://planetmath.org/IndexingSet

) are all 0 and that

sgn

⁢

(

E

2

⁢

m

)

=

(

-

1

)

m

for

m

=

0

,

1

,

2

,

…

The Euler number have intimate relation to the

Bernoulli numbers

.  The first Euler numbers with even index are

E

0

=

1

,

E

2

=

-

1

,

E

4

=

5

,

E

6

=

-

61

,

E

8

=

1385

,

E

10

=

-

50521

.

•

One can by hand determine Euler numbers by performing the

division

of 1 by the

Taylor series

of

hyperbolic cosine

(cf.

Taylor series via division

and

Taylor series of hyperbolic
functions

).
Since

cosh

⁡

i

⁢

x

=

cos

⁡

x

,  the division

1

:

cos

⁡

x

correspondingly gives only terms with plus sign, i.e. it shows the

absolute values

of the Euler numbers.

•

The Euler numbers may also be obtained by using the

Euler polynomials

E

n

⁢

(

x

)

:

E

n

=

2

n

⁢

E

n

⁢

(

1

2

)

•

If the Euler numbers

E

k

are denoted as symbolic powers

E

k

, then one may write the equation

(

E

+

1

)

n

+

(

E

-

1

)

n

=

0

,

which can be used as a

recurrence relation

for computing the values of the even-indexed Euler numbers.  Cf. the

Leibniz rule

for

derivatives

of product

f

⁢

g

.

Title

Euler numbers

Canonical name

EulerNumbers

Date of creation

2014-12-02 17:43:40

Last modified on

2014-12-02 17:43:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 11B68

Related topic

GudermannianFunction

Related topic

BernoulliNumber

Related topic

InverseGudermannianFunction

Related topic

HermiteNumbers

Generated on Fri Feb 9 15:29:13 2018 by

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Euler polynomial

Euler polynomial

The

Euler polynomials

E

0

⁢

(

x

)

,

E

1

⁢

(

x

)

,

E

2

⁢

(

x

)

,

…

are certain

polynomials

of the indeterminate

x

with

rational

coefficients (whose

denominators

may only be powers

1

,

2

,

4

,

8

,

…

of 2).  The Euler polynomials may be defined by means of the

generating function

such that

2

⁢

e

x

⁢

t

e

t

+

1

=

∑

n

=

0

∞

E

n

⁢

(

x

)

⁢

t

n

n

!

,

i.e. one can get them by dividing the

Taylor series

2

+

2

⁢

x

⁢

t

+

x

2

⁢

t

2

+

1

3

⁢

x

3

⁢

t

3

+

…

by the Taylor series

2

+

t

+

1

2

⁢

t

2

+

1

6

⁢

t

3

+

…

.  There are also explicit formulae for the polynomials, e.g.

E

n

⁢

(

x

)

=

∑

k

=

0

n

(

n

k

)

⁢

E

k

2

k

⁢

(

x

-

1

2

)

n

-

k

via the

Euler numbers

E

k

.

Conversely

, the Euler numbers are expressed with the Euler polynomials through

E

k

=

2

k

⁢

E

k

⁢

(

1

2

)

.

The first seven Euler polynomials are

E

0

⁢

(

x

)

=

1

E

1

⁢

(

x

)

=

x

-

1

2

E

2

⁢

(

x

)

=

x

2

-

x

E

3

⁢

(

x

)

=

x

3

-

3

2

⁢

x

2

+

1

4

E

4

⁢

(

x

)

=

x

4

-

2

⁢

x

3

+

x

E

5

⁢

(

x

)

=

x

5

-

5

2

⁢

x

4

+

5

2

⁢

x

2

-

1

2

E

6

⁢

(

x

)

=

x

6

-

3

⁢

x

5

+

5

⁢

x

3

-

3

⁢

x

The Euler polynomials have the beautiful

addition formula

E

n

⁢

(

x

+

y

)

=

∑

k

=

0

n

(

n

k

)

⁢

E

k

⁢

(

x

)

⁢

y

k

and the

derivative

E

n

′

(

x

)

=

n

E

n

-

1

(

x

)

(

for

n

=

1

,

2

,

…

)

.

The Euler polynomials form an example of

Appell sequences

.

Title

Euler polynomial

Canonical name

EulerPolynomial

Date of creation

2013-03-22 19:07:07

Last modified on

2013-03-22 19:07:07

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 11B68

Related topic

BernoulliPolynomial

Related topic

BernoulliPolynomialsAndNumbers

Generated on Fri Feb 9 15:29:17 2018 by

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generalized Bernoulli number

generalized Bernoulli number

Let

χ

be a non-trivial

primitive character

mod

m

. The

generalized Bernoulli numbers

B

n

,

χ

are given by

∑

a

=

1

m

χ

⁢

(

a

)

⁢

t

⁢

e

a

⁢

t

e

m

⁢

t

-

1

=

∑

n

=

0

∞

B

n

,

χ

⁢

t

n

n

!

They are members of the field

ℚ

⁢

(

χ

)

generated by the values of

χ

.

Title

generalized Bernoulli number

Canonical name

GeneralizedBernoulliNumber

Date of creation

2013-03-22 13:22:40

Last modified on

2013-03-22 13:22:40

Owner

sucrose (1410)

Last modified by

sucrose (1410)

Numerical id

6

Author

sucrose (1410)

Entry type

Definition

Classification

msc 11B68

Related topic

BernoulliNumber

Generated on Fri Feb 9 15:29:19 2018 by

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Hermite numbers

Hermite numbers

The

Hermite numbers

H

n

may be defined by the generating function

e

-

t

2

:=

∑

n

=

0

∞

H

n

⁢

t

n

n

!

(1)

which is same as the

generating function of Hermite polynomials

at the value 0 of the

argument

z

.  After expanding the left hand side of (1) to

Taylor series

1

-

t

2

1

!

+

t

4

2

!

-

t

6

3

!

+

-

…

, one can write

1

-

2

⋅

t

2

2

!

+

12

⋅

t

4

4

!

-

120

⋅

t

6

6

!

+

-

…

=

∑

n

=

0

∞

H

n

t

n

n

!

.

(2)

Thus one sees that

H

0

=

1

,

H

1

=

0

,

H

2

=

-

2

,

H

3

=

0

,

H

4

=

12

,

H

5

=

0

,

H

6

=

-

120

,

…

Evidently,

H

n

=

{

(

-

1

)

n

2

⁢

n

!

(

n

2

)

!

when

2

∣

n

0

when

⁢

2

∤

n

(3)

The Hermite numbers form the

sequence

(

http://www.research.att.com/ njas/sequences/index.html?q=A067994&language=english&go=Search

Sloane A067994)

1

,

0

,

-

2

,

0

,

12

,

0

,

-

120

,

0

,

1680

,

0

,

-

30240

,

0

,

665280

,

0

,

-

17297280

,

0

,

…

which obeys the

recurrence relation

H

n

=

2

⁢

(

1

-

n

)

⁢

H

n

-

2

.

(4)

According to (1), the Hermite numbers satisfy

H

n

=

H

n

⁢

(

0

)

where

H

n

⁢

(

x

)

is the

Hermite polynomial

of degree

n

.  The of Hermite numbers and Hermite polynomials may be expressed also by using symbolic powers

H

ν

=

:

H

ν

as follows:

(

2

⁢

x

+

H

)

n

=

H

n

⁢

(

x

)

.

(5)

This means e.g. that

(

2

⁢

x

+

H

)

2

=

(

2

⁢

x

)

2

+

2

⋅

2

⁢

x

⁢

H

1

+

H

2

=

4

⁢

x

2

+

4

⁢

x

⁢

H

1

+

H

2

=

4

⁢

x

2

-

2

=

H

2

⁢

(

x

)

.

Title

Hermite numbers

Canonical name

HermiteNumbers

Date of creation

2013-03-22 19:08:32

Last modified on

2013-03-22 19:08:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 11B68

Related topic

EulerNumbers2

Related topic

AppellSequence

Generated on Fri Feb 9 15:29:21 2018 by

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Kummer’s congruence

Kummer’s congruence

Let

B

k

denote the

k

th

Bernoulli number

:

B

0

=

1

,

B

1

=

-

1

2

,

B

2

=

1

6

,

B

3

=

0

,

B

4

=

-

1

30

,

…

,

B

10

=

5

66

,

…

In fact,

B

k

=

0

for all odd

k

≥

3

, so we will only consider

B

k

for even

k

. The following

congruence

is due to Ernst Eduard Kummer:

Theorem

(Kummer’s congruence).

Let

p

be a prime. Suppose that

k

≥

2

is an

even integer

which is not

divisible

by

(

p

-

1

)

. Then the quotient

B

k

/

k

is

p

-integral, that is, as a

fraction

in lower terms,

p

does not divide its denominator. Furthermore, if

h

is another even integer with

(

p

-

1

)

∤

k

and

k

≡

h

mod

(

p

-

1

)

then

B

k

k

≡

B

h

h

mod

p

.

The interested reader should see also the

congruence of Clausen and von Staudt

for a

similar

result. As an example of Kummer’s congruence, let

p

=

7

and

k

=

4

. Then:

B

4

4

=

-

1

30

4

=

-

1

120

≡

6

mod

7

If we pick

h

=

10

(so that

10

≡

4

mod

(

p

-

1

)

) then:

B

10

10

=

5

66

10

=

1

132

≡

6

mod

7

which is what the

theorem

predicted.

Title

Kummer’s congruence

Canonical name

KummersCongruence

Date of creation

2013-03-22 15:12:01

Last modified on

2013-03-22 15:12:01

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11B68

Synonym

Kummer congruence

Related topic

CongruenceOfClausenAndVonStaudt

Related topic

IntegralElement

Related topic

OddBernoulliNumbersAreZero

Generated on Fri Feb 9 15:29:24 2018 by

LaTeXML
recurrence formula for Bernoulli numbers

recurrence formula for Bernoulli numbers

This article establishes a well-known recurrence formula for the

Bernoulli numbers

.

The

Bernoulli polynomials

b

r

⁢

(

x

)

,

r

≥

1

can be written explicitly as

b

r

⁢

(

x

)

=

∑

k

=

1

r

(

r

k

)

⁢

B

r

-

k

⁢

x

k

+

B

r

(see this article (

http://planetmath.org/CoefficientsOfBernoulliPolynomials

)).

For

r

≥

2

, we have

0

=

∫

0

1

b

r

-

1

(

x

)

d

x

=

1

r

b

r

(

x

)

|

0

1

=

1

r

(

b

r

(

1

)

-

b

r

(

0

)

)

and thus

B

r

=

b

r

⁢

(

0

)

=

b

r

⁢

(

1

)

=

∑

k

=

1

r

(

r

k

)

⁢

B

r

-

k

+

B

r

It follows that (still when

r

≥

2

)

∑

k

=

1

r

(

r

k

)

⁢

B

r

-

k

=

0

so that

(

r

1

)

⁢

B

r

-

1

=

-

∑

k

=

2

r

(

r

k

)

⁢

B

r

-

k

Replacing

r

by

r

+

1

and simplifying, we see that for

r

≥

1

,

B

r

=

-

1

r

+

1

⁢

∑

k

=

2

r

+

1

(

r

+

1

k

)

⁢

B

r

+

1

-

k

=

-

1

r

+

1

⁢

∑

k

=

1

r

(

r

+

1

k

+

1

)

⁢

B

r

-

k

Title

recurrence formula for Bernoulli numbers

Canonical name

RecurrenceFormulaForBernoulliNumbers

Date of creation

2013-03-22 17:46:19

Last modified on

2013-03-22 17:46:19

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Derivation

Classification

msc 11B68

Generated on Fri Feb 9 15:29:38 2018 by

LaTeXML
the odd Bernoulli numbers are zero

the odd Bernoulli numbers are zero

Recall that, for

k

≥

0

, the

Bernoulli numbers

B

k

are defined as the

coefficients

in the

Taylor expansion

:

t

e

t

-

1

=

∑

k

≥

0

B

k

⁢

t

k

k

!

.

(1)

Just to name a few:

B

0

=

1

,

B

1

=

-

1

2

,

B

2

=

1

6

,

B

3

=

0

,

B

4

=

-

1

30

,

B

5

=

0

,

…

,

B

10

=

5

66

,

…

Lemma

.

If

k

≥

3

is odd then

B

k

=

0

.

Proof.

From the right hand side of (

1

) we extract the term corresponding to

k

=

1

:

t

e

t

-

1

=

-

t

2

+

∑

k

≥

0

,

k

≠

1

B

k

⁢

t

k

k

!

.

(2)

Thus:

t

e

t

-

1

+

t

2

=

∑

k

≥

0

,

k

≠

1

B

k

⁢

t

k

k

!

(3)

and the left hand side can be rewritten as:

t

e

t

-

1

+

t

2

=

2

⁢

t

+

t

⁢

(

e

t

-

1

)

2

⁢

(

e

t

-

1

)

=

t

2

⋅

e

t

+

1

e

t

-

1

=

t

2

⋅

e

t

/

2

+

e

-

t

/

2

e

t

/

2

-

e

-

t

/

2

.

(4)

Hence, if one replaces

t

by

-

t

then (

4

) is unchanged. Since (

4

) is the left hand side of (

3

), the quantity

∑

k

≥

0

,

k

≠

1

B

k

⁢

t

k

k

!

is also unchanged when

t

is exchanged by

-

t

, and so we must have

B

k

=

(

-

1

)

k

⁢

B

k

for

k

≠

1

. We conclude that if

k

≥

3

and

k

is odd,

B

k

=

0

.
∎

Title

the

odd Bernoulli numbers are zero

Canonical name

TheOddBernoulliNumbersAreZero

Date of creation

2013-03-22 15:12:04

Last modified on

2013-03-22 15:12:04

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11B68

Related topic

KummersCongruence

Related topic

CongruenceOfClausenAndVonStaudt

Generated on Fri Feb 9 15:29:46 2018 by

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Erdős-Heilbronn conjecture

Erdős-Heilbronn conjecture

Let

A

⊂

ℤ

p

be a set of

residues

modulo

p

, and let

h

be a

positive integer

, then

h

∧

⁢

A

=

{

a

1

+

a

2

+

⋯

+

a

h

∣

a

1

,

a

2

,

…

,

a

h

⁢

are distinct elements of

⁢

A

}

has cardinality at least

min

⁡

(

p

,

h

⁢

k

-

h

2

+

1

)

. This was conjectured by Erdős and Heilbronn in 1964

[

1

]

. The first proof was given by Dias da Silva and Hamidoune in 1994.

References

1

Paul Erdős and Hans Heilbronn.

On the

addition

of

residue classes

mod

p

.

Acta Arith.

, 9:149–159, 1964.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0156.04801

Zbl 0156.04801.

2

Melvyn B. Nathanson.

Additive Number Theory

:

Inverse Problems

and Geometry of
Sumsets

, volume 165 of

GTM

.

Springer, 1996.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003

Zbl 0859.11003.

Title

Erdős-Heilbronn conjecture

Canonical name

ErdHosHeilbronnConjecture

Date of creation

2013-03-22 13:38:15

Last modified on

2013-03-22 13:38:15

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

10

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11B75

Related topic

CauchyDavenportTheorem

Generated on Fri Feb 9 15:29:48 2018 by

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Freiman isomorphism

Freiman isomorphism

Let

A

and

B

be subsets of

abelian groups

G

A

and

G

B

respectively. A

Freiman isomorphism

of order

s

is a

bijective

mapping

f

:

A

→

B

such that

a

1

+

a

2

+

⋯

+

a

s

=

a

1

′

+

a

2

′

+

⋯

+

a

s

′

holds if and only if

f

⁢

(

a

1

)

+

f

⁢

(

a

2

)

+

⋯

+

f

⁢

(

a

s

)

=

f

⁢

(

a

1

′

)

+

f

⁢

(

a

2

′

)

+

⋯

+

f

⁢

(

a

s

′

)

.

The Freiman isomorphism is a

restriction

of the conventional notion of
a

group isomorphism

to a limited number of

group operations

. In
particular, a Freiman isomorphism of order

s

is also a Freiman
isomorphism of order

s

-

1

, and the mapping is a Freiman isomorphism of
every order precisely when it is the conventional

isomorphism

.

Freiman isomorphisms were introduced by Freiman in his monograph

[

1

]

to build a general

theory of
set

addition

(

http://planetmath.org/Sumset

) that is

independent

of the
underlying group.

The number of

equivalence classes

of

n

-

element

sets of integers under Freiman isomorphisms of order

2

is

n

2

⁢

n

⁢

(

1

+

o

⁢

(

1

)

)

[

2

]

.

References

1

Gregory Freiman.

Foundations

of Structural Theory of Set Addition

, volume 37 of

Translations

of Mathematical Monographs

.

AMS, 1973.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0271.10044

Zbl
0271.10044.

2

Sergei V. Konyagin and Vsevolod F. Lev.

Combinatorics

and

linear algebra

of Freiman’s isomorphism.

Mathematika

, 47:39–51, 2000.

Available at

http://math.haifa.ac.il/ seva/pub_list.html

http://math.haifa.ac.il/ seva/.

3

Melvyn B. Nathanson.

Additive Number Theory

:

Inverse Problems

and Geometry of
Sumsets

, volume 165 of

GTM

.

Springer, 1996.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003

Zbl 0859.11003.

Title

Freiman isomorphism

Canonical name

FreimanIsomorphism

Date of creation

2013-03-22 13:40:39

Last modified on

2013-03-22 13:40:39

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B75

Classification

msc 20K30

Related topic

Isomorphism2

Generated on Fri Feb 9 15:29:50 2018 by

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sum-free

sum-free

A set

A

is called

sum-free

if the equation

a

1

+

a

2

=

a

3

does not have solutions in elements of

A

. Equivalently, a set is sum-free if it is disjoint from its

2

-fold sumset, i.e.,

A

∩

2

⁢

A

=

∅

.

Title

sum-free

Canonical

name

Sumfree

Date of creation

2013-03-22 13:39:08

Last modified on

2013-03-22 13:39:08

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

5

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B75

Generated on Fri Feb 9 15:29:51 2018 by

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Beatty sequence

Beatty sequence

The

integer sequence

ℬ

⁢

(

α

,

α

′

)

:=

(

⌊

n

-

α

′

α

⌋

)

n

=

1

∞

is called the

Beatty sequence

with density

α

, slope

1

α

, offset

α

′

, and

y

-intercept

-

α

′

α

.

Sometimes a

sequence

of the above type is called a

floor

Beatty sequence, and denoted

ℬ

(

f

)

⁢

(

α

,

α

′

)

, while an integer sequence

ℬ

(

c

)

⁢

(

α

,

α

′

)

:=

(

⌈

n

-

α

′

α

⌉

)

n

=

1

∞

is called a

ceiling

Beatty sequence.

References

[1

] M. Lothaire, , vol. 90, Cambridge University Press, Cambridge, 2002, ISBN 0-521-81220-8, available online at

http://www-igm.univ-mlv.fr/~berstel/Lothaire

. A collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio,
Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique
Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo,
Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques
Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and
Wojciech Plandowski;
With a preface by Berstel and Perrin.

http://www.ams.org/mathscinet-getitem?mr=1905123

MR 1905123

Title

Beatty sequence

Canonical name

BeattySequence

Date of creation

2013-03-22 13:37:23

Last modified on

2013-03-22 13:37:23

Owner

Kevin OBryant (1315)

Last modified by

Kevin OBryant (1315)

Numerical id

12

Author

Kevin OBryant (1315)

Entry type

Definition

Classification

msc 11B83

Related topic

BeattysTheorem

Related topic

FraenkelsPartitionTheorem

Related topic

DataStream

Related topic

WideraInterlaceAndDeinterlace

Generated on Fri Feb 9 15:29:53 2018 by

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Beatty’s theorem

Beatty’s theorem

If

p

and

q

are

positive

irrationals

such that

1

p

+

1

q

=

1

then the

sequences

{

⌊

n

p

⌋

}

n

=

1

∞

=

⌊

p

⌋

,

⌊

2

⁢

p

⌋

,

⌊

3

⁢

p

⌋

,

…

{

⌊

n

q

⌋

}

n

=

1

∞

=

⌊

q

⌋

,

⌊

2

⁢

q

⌋

,

⌊

3

⁢

q

⌋

,

…

where

⌊

x

⌋

denotes the floor (or

greatest integer function

) of

x

, constitute a

partition

of the set of positive integers.

That is, every

positive integer

is a member exactly once of one of the two sequences and the two sequences have no common terms.

Title

Beatty’s theorem

Canonical name

BeattysTheorem

Date of creation

2013-03-22 11:56:34

Last modified on

2013-03-22 11:56:34

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Theorem

Classification

msc 11B83

Related topic

Sequence

Related topic

Irrational

Related topic

Partition

Related topic

Floor

Related topic

Ceiling

Related topic

BeattySequence

Related topic

FraenkelsPartitionTheorem

Related topic

FraenkelsPartitionTheorem2

Related topic

ConjugateIndex

Generated on Fri Feb 9 15:29:56 2018 by

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data stream

data stream

A

data stream

is an

ordered pair

(

s

,

Δ

)

where:

1.

s

is a sequence of tuples,

2.

Δ

is the

sequence

of time intervals (i.e.

rational

or real numbers) and each

Δ

n

>

0

Title

data stream

Canonical name

DataStream

Date of creation

2013-03-22 15:37:06

Last modified on

2013-03-22 15:37:06

Owner

michal (7107)

Last modified by

michal (7107)

Numerical id

11

Author

michal (7107)

Entry type

Definition

Classification

msc 11B83

Synonym

time series

Synonym

sequence of tuples

Synonym

stream

Related topic

BeattySequence

Related topic

FraenkelsPartitionTheorem

Related topic

WideraInterlaceAndDeinterlace

Defines

Stream of Tuples

Generated on Fri Feb 9 15:29:57 2018 by

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Fraenkel’s partition theorem

Fraenkel’s partition theorem

Fraenkel’s

partition

theorem

is a

generalization

of Beatty’s Theorem. Set

ℬ

⁢

(

α

,

α

′

)

:=

(

⌊

n

-

α

′

α

⌋

)

n

=

1

∞

.

We say that two

sequences

partition

ℕ

=

{

1

,

2

,

3

,

…

}

if the sequences are

disjoint

and their union is

ℕ

.

Fraenkel’s Partition Theorem:

The sequences

B

⁢

(

α

,

α

′

)

and

B

⁢

(

β

,

β

′

)

partition

N

if and only if the following five conditions are
satisfied.

1.

0

<

α

<

1

.

2.

α

+

β

=

1

.

3.

0

≤

α

+

α

′

≤

1

.

4.

If

α

is

irrational

, then

α

′

+

β

′

=

0

and

k

⁢

α

+

α

′

∉

ℤ

for

2

≤

k

∈

ℕ

.

5.

If

α

is

rational

(say

q

∈

ℕ

is

minimal

with

q

⁢

α

∈

ℕ

), then

1

q

≤

α

+

α

′

and

⌈

q

⁢

α

′

⌉

+

⌈

q

⁢

β

′

⌉

=

1

.

References

[1

] Aviezri S. Fraenkel,

The

bracket function

and

complementary

sets of integers

, Canad. J.
Math.

21

(1969), 6–27.

http://www.ams.org/mathscinet-getitem?mr=38:3214

MR
38:3214

[2

] Kevin O’Bryant,

Fraenkel’s partition and Brown’s decomposition

,

http://lanl.arxiv.org/abs/math.NT/0305133

arXiv:math.NT/0305133.

Title

Fraenkel’s partition theorem

Canonical name

FraenkelsPartitionTheorem

Date of creation

2013-03-22 13:40:09

Last modified on

2013-03-22 13:40:09

Owner

Kevin OBryant (1315)

Last modified by

Kevin OBryant (1315)

Numerical id

6

Author

Kevin OBryant (1315)

Entry type

Theorem

Classification

msc 11B83

Synonym

Fraenkel’s theorem

Related topic

BeattySequence

Related topic

BeattysTheorem

Related topic

DataStream

Related topic

WideraInterlaceAndDeinterlace

Generated on Fri Feb 9 15:29:59 2018 by

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