pentagonal number theorem

pentagonal number theorem

Theorem

:

∏

k

=

1

∞

(

1

-

x

k

)

=

∑

n

=

-

∞

∞

(

-

1

)

n

⁢

x

n

⁢

(

3

⁢

n

+

1

)

/

2

(1)

where the two sides are regarded as formal power series over

ℤ

.

Proof:

For

n

≥

0

, denote by

f

⁢

(

n

)

the coefficient of

x

n

in the

product

on the left, i.e. write

∏

k

=

1

∞

(

1

-

x

k

)

=

∑

n

=

0

∞

f

⁢

(

n

)

⁢

x

n

.

By this definition, we have for all

n

f

⁢

(

n

)

=

e

⁢

(

n

)

-

d

⁢

(

n

)

where

e

⁢

(

n

)

(resp.

d

⁢

(

n

)

) is the number of

partitions

of

n

as
a sum of an even (resp. odd) number of distinct summands.
To fix the notation, let

P

⁢

(

n

)

be set of pairs

(

s

,

g

)

where

s

is
a

natural number

>

0

and

g

is a decreasing

mapping

{

1

,

2

,

…

,

s

}

→

ℕ

+

such that

∑

x

g

⁢

(

x

)

=

n

.
The

cardinal

of

P

⁢

(

n

)

is thus

f

⁢

(

n

)

, and

P

⁢

(

n

)

is the union of
these two

disjoint sets

:

E

⁢

(

n

)

=

{

(

s

,

g

)

∈

P

⁢

(

n

)

∣

s

⁢

is even

}

,

D

⁢

(

n

)

=

{

(

s

,

g

)

∈

P

⁢

(

n

)

∣

s

⁢

is odd

}

.

Now on the right side of (

1

) we have

1

+

∑

n

=

1

∞

(

-

1

)

n

⁢

x

n

⁢

(

3

⁢

n

+

1

)

/

2

+

∑

n

=

1

∞

(

-

1

)

n

⁢

x

n

⁢

(

3

⁢

n

-

1

)

/

2

.

Therefore what we want to prove is

e

⁢

(

n

)

=

d

⁢

(

n

)

+

(

-

1

)

m

if

⁢

n

=

m

⁢

(

3

⁢

m

±

1

)

/

2

⁢

for some

⁢

m

(2)

e

⁢

(

n

)

=

d

⁢

(

n

)

otherwise.

(3)

For

m

≥

1

we have

m

⁢

(

3

⁢

m

+

1

)

/

2

=

2

⁢

m

+

(

2

⁢

m

-

1

)

+

…

+

(

m

+

1

)

(4)

m

⁢

(

3

⁢

m

-

1

)

/

2

=

(

2

⁢

m

-

1

)

+

(

2

⁢

m

-

2

)

+

…

+

m

(5)

Take some

(

s

,

g

)

∈

P

⁢

(

n

)

, and suppose first that

n

is not
of the form (

4

) nor (

5

).
Since

g

is decreasing, there is a unique

integer

k

∈

[

1

,

s

]

such that

g

⁢

(

j

)

=

g

⁢

(

1

)

-

j

+

1

for

⁢

j

∈

[

1

,

k

]

,

g

⁢

(

j

)

<

g

⁢

(

1

)

-

j

+

1

for

⁢

j

∈

[

k

+

1

,

s

]

.

If

g

⁢

(

s

)

≤

k

, define

g

¯

:

[

1

,

s

-

1

]

→

ℕ

+

by

g

¯

⁢

(

x

)

=

{

g

⁢

(

x

)

+

1

,

if

⁢

x

∈

[

1

,

g

⁢

(

s

)

]

,

g

⁢

(

x

)

,

if

⁢

x

∈

[

g

⁢

(

s

)

+

1

,

s

-

1

]

.

If

g

⁢

(

s

)

>

k

, define

g

¯

:

[

1

,

s

+

1

]

→

ℕ

+

by

g

¯

⁢

(

x

)

=

{

g

⁢

(

x

)

-

1

,

if

⁢

x

∈

[

1

,

k

]

,

g

⁢

(

x

)

,

if

⁢

x

∈

[

k

+

1

,

s

]

,

k

,

if

⁢

x

=

s

+

1

.

In both cases,

g

¯

is decreasing and

∑

x

g

¯

⁢

(

x

)

=

n

.
The mapping

g

→

g

¯

maps takes an

element

having odd

s

to an element having even

s

, and vice versa.
Finally, the reader can verify that

g

¯

¯

=

g

.
Thus we have constructed a

bijection

E

⁢

(

n

)

→

D

⁢

(

n

)

,
proving (

3

).

Now suppose that

n

=

m

⁢

(

3

⁢

m

+

1

)

/

2

for some (perforce unique)

m

.
The above construction still yields a bijection between

E

⁢

(

n

)

and

D

⁢

(

n

)

excluding (from one set or the other) the single
element

(

m

,

g

0

)

:

g

0

⁢

(

x

)

=

2

⁢

m

+

1

-

x

for

⁢

x

∈

[

1

,

m

]

as in (

4

).
Likewise if

n

=

m

⁢

(

3

⁢

m

-

1

)

/

2

, only this element

(

m

,

g

1

)

is excluded:

g

1

⁢

(

x

)

=

2

⁢

m

-

x

for

⁢

x

∈

[

1

,

m

]

as in (

5

).
In both cases we deduce (

2

), completing the proof.

Remarks:

The name of the theorem derives from the fact that
the

exponents

n

⁢

(

3

⁢

n

+

1

)

/

2

are the generalized

pentagonal numbers

.

The theorem was discovered and proved by Euler around 1750.
This was one of the first results about what are now called

theta
functions

, and was also one of the earliest applications of
the

formalism

of

generating functions

.

The above proof is due to F. Franklin,
(

Comptes Rendus de l’Acad. des Sciences

,

92

, 1881, pp. 448-450).

Title

pentagonal number theorem

Canonical name

PentagonalNumberTheorem

Date of creation

2013-03-22 13:57:51

Last modified on

2013-03-22 13:57:51

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11P81

Classification

msc 14K25

Generated on Fri Feb 9 15:40:34 2018 by

LaTeXML