Classical Partition Function and Euler’s Product Formula

Abstract

With various Substitution of q and z, Jacobi’s triple product identity gives many interesting results. For example, if we put q = q^3/2 and then z = −q^−1/2 into (11.1), we get

$$ \sum\limits_{n \in \mathbb{Z}} {( - 1)^n q^{\frac{{(3n^2 - n)}} {2}} } = \prod\limits_{n = 1}^\infty {(1 - q^{3n} )(1 - q^{3n - 2} )} (1 - q^{3n - 1} ) = \prod\limits_{n = 1}^\infty {(1 - q^n ),} $$

((12.1))

which is called Euler’s product formula. We proved that it holds when |q| < 1. It follows that it also holds as an equality of formal power series in q (see Chapter 8). The formula may also be written using Euler’s product

$$ \phi (q) = \prod\limits_{n = 1}^\infty {(1 - q^n )} $$

as

$$ \phi (q) = \sum\limits_{n \in \mathbb{Z}} {( - 1)^n q^{e_n } } , $$

((12.2))

where

$$ e_n = \frac{{3n^2 - n}} {2} $$

((12.3))

are called pentagonal numbers. The reader is encouraged to multiply out the first few factors of Euler’s product to discover the astonishing fact that indeed the enth coefficient is (−1)ⁿ and all other coefficients are zero.