Extensions of an identity of Chan and Cooper in the spirit of Ramanujan

Abstract

Chan and Cooper proved that if the integers \({c(n) (n=0,1,2,\ldots )}\) are given by

$$\begin{aligned} \sum _{n=0}^\infty c(n)q^n = \prod _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^2\left( 1-q^{3n}\right) ^2}, \end{aligned}$$

then

$$\begin{aligned} \sum _{n=0}^\infty c(2n+1)q^n = 2 \prod _{n=1}^\infty \frac{\left( 1-q^{2n}\right) ^4\left( 1-q^{6n}\right) ^4}{\left( 1-q^n\right) ^6\left( 1-q^{3n}\right) ^6}. \end{aligned}$$

We prove many other results of this type and apply them to the determination of congruence properties of the coefficients.