Jacobi triple product

In mathematics, the Jacobi triple product is the identity:

∏

m

=

1

∞

(

1

−

x

2

m

)

(

1

+

x

2

m

−

1

y

2

)

(

1

+

x

2

m

−

1

y

2

)

=

∑

n

=

−

∞

∞

x

n

2

y

2

n

,

{\displaystyle \prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+{\frac {x^{2m-1}}{y^{2}}}\right)=\sum _{n=-\infty }^{\infty }x^{n^{2}}y^{2n},}

for complex numbers x and y, with |x| < 1 and y ≠ 0. It was introduced by Jacobi (1829) in his work Fundamenta Nova Theoriae Functionum Ellipticarum.

The Jacobi triple product identity is the Macdonald identity for the affine root system of type A1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra.

Properties

Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi triple product identity.

Let

x

=

q

q

{\displaystyle x=q{\sqrt {q}}}

and

y

2

=

−

q

{\displaystyle y^{2}=-{\sqrt {q}}}

. Then we have

ϕ

(

q

)

=

∏

m

=

1

∞

(

1

−

q

m

)

=

∑

n

=

−

∞

∞

(

−

1

)

n

q

3

n

2

−

n

2

.

{\displaystyle \phi (q)=\prod _{m=1}^{\infty }\left(1-q^{m}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{\frac {3n^{2}-n}{2}}.}

The Rogers–Ramanujan identities follow with

x

=

q

2

q

{\displaystyle x=q^{2}{\sqrt {q}}}

,

y

2

=

−

q

{\displaystyle y^{2}=-{\sqrt {q}}}

and

x

=

q

2

q

{\displaystyle x=q^{2}{\sqrt {q}}}

,

y

2

=

−

q

q

{\displaystyle y^{2}=-q{\sqrt {q}}}

.

The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:

Let

x

=

e

i

π

τ

{\displaystyle x=e^{i\pi \tau }}

and

y

=

e

i

π

z

.

{\displaystyle y=e^{i\pi z}.}

Then the Jacobi theta function

ϑ

(

z

;

τ

)

=

∑

n

=

−

∞

∞

e

π

i

n

2

τ

+

2

π

i

n

z

{\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }e^{\pi {\rm {i}}n^{2}\tau +2\pi {\rm {i}}nz}}

can be written in the form

∑

n

=

−

∞

∞

y

2

n

x

n

2

.

{\displaystyle \sum _{n=-\infty }^{\infty }y^{2n}x^{n^{2}}.}

Using the Jacobi triple product identity, the theta function can be written as the product

ϑ

(

z

;

τ

)

=

∏

m

=

1

∞

(

1

−

e

2

m

π

i

τ

)

[

1

+

e

(

2

m

−

1

)

π

i

τ

+

2

π

i

z

]

[

1

+

e

(

2

m

−

1

)

π

i

τ

−

2

π

i

z

]

.