Quintuple product identity

Statement

Let $s, t \in \C$ with $\abs{s} < 1$ and $t \ne 0$ . Then $\prod_{n=1}^{\infty} \left(1 - s^n\right) \left(1 - s^n t\right) \left(1 - s^{n-1} t^{-1}\right) \left(1 - s^{2n-1} t^2\right) \left(1 - s^{2n-1} t^{-2}\right) = \sum_{n=-\infty}^{\infty} s^{(3n^2 + n)/2} \left( t^{3n} - t^{-3n-1} \right).$

Remarks

Both sides are holomorphic in $t$ on $\C \setminus \set{0}$ , and the infinite product converges because $\abs{s} < 1$ . Proof idea: as for the Jacobi triple product, one checks that both sides $F$ satisfy $F(st) = s^{-2} t^{-3} F(t)$ . This relation links the Laurent coefficients of index $k$ and $k + 3$ , so it fixes them from three of them, and these are compared directly. Example: at $t = 1$ the factor $1 - s^0 t^{-1}$ of index $n = 1$ vanishes, so the left side is $0$ . On the right the terms $s^{(3n^2+n)/2}(1 - 1)$ also vanish, so both sides agree. The exponents $(3n^2 + n)/2$ are the generalized pentagonal numbers, as in Euler's pentagonal number theorem.

Going deeper

For a student

The identity has been rediscovered many times, and there are many equivalent forms obtained by changes of variables. It can be derived from the Jacobi triple product, and it is a source of identities for theta functions and for the Dedekind eta function.

Reading path

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Complex numbers object
Sequence definition
Infinite product definition
Quintuple product identity statement

Tag	`PZR6RC79ED`
URI	`m2://PZR6RC79ED`
Kind	statement, theorem
Writing	W0 stub
MSC	11F27, 33D15
Status	This copy of the site has no build data. Run `m2 status PZR6RC79ED`.
About	Theta function
Uses	Infinite product
Same as	Wikidata Q7272511 Wikipedia Quintuple product identity