Jacobi triple product

For

∣ q ∣ < 1

and

z \neq = 0

, the Jacobi triple product is

\sum_{n \in Z} z^{n} q^{n^{2}} = \prod_{m \geq 1} (1 - q^{2 m}) (1 + z q^{2 m - 1}) (1 + z^{- 1} q^{2 m - 1}) .

Table of contents
- Jacobi cubic product identity Jacobi triple product
- Euler pentagonal identity Jacobi triple product

Jacobi cubic product identity ( $\prod_{n \geq 1} (1 - q^{n})^{3} = \sum_{r \geq 0} (- 1)^{r} (2 r + 1) q^{r (r + 1) /2}$ )

The odd Dirichlet character modulo four has

χ_{4} (2 r + 1) = (- 1)^{r}

B = \frac{1}{2} \sum_{n} n χ_{4} (n) q^{n^{2} /8}

is therefore

q^{1/8}

times the displayed sum. The character theta proof of the eta product gives

B = η^{3}

, proving the identity.

Euler pentagonal identity ( $\prod_{n \geq 1} (1 - q^{n}) = \sum_{r \in Z} (- 1)^{r} q^{r (3 r + 1) /2}$ )

The identity holds for

∣ q ∣ < 1

, with both sides locally uniformly convergent. In the character theta proof of the eta product, pair the terms indexed by

n

and

- n

and then select

n = 6 r + 1

; their Dirichlet character signs are

(- 1)^{r}

. Removing

q^{1/24}

gives the displayed product identity.

 Ancestors (8)

 Incoming links (7)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Jacobi triple product by

Wikipedia Bot  1

The Jacobi triple product is an important identity in the theory of partitions and combinatorial mathematics. It relates the series expansion of certain infinite products and has applications in number theory, combinatorics, and the study of special functions.

 Read the full article

  See all articles in the same topic Create my own version