Algebraic relations between certain infinite products

Proceedings of the London Mathematical Society, 2, XVIII, 1920, Records for 13 March 1919

It was proved by Prof. L. J. Rogers^{1} that
\begin{align*}
G(x)&= 1+\frac{1}{1-x}+\frac{x^4}{(1-x)(1-x^2)}+
\frac{x^9}{(1-x)(1-x^2)(1-x^3)}+\cdots\\
&= \frac{1}{(1-x)(1-x^6)(1-x^{11})}\cdots \times
\frac{1}{(1-x^4)(1-x^9)(1-x^{14})\cdots},
\end{align*}
and
\begin{align*}
H(x)&= 1+\frac{x^2}{1-x}+\frac{x^6}{(1-x)(1-x^2)}+
\frac{x^{12}}{(1-x)(1-x^2)(1-x^3)}+\cdots\\
&= \frac{1}{(1-x^2)(1-x^7)(1-x^{12})}\cdots \times
\frac{1}{(1-x^3)(1-x^8)(1-x^{13})\cdots}.
\end{align*}
Simpler proofs were afterwards found Prof. Rogers and myself.^{2}

I have now found an algebraic relation between $G(x)$ and $H(x)$, viz.: $$H(x)\{G(x)\}^{11}-x^2G(x)\{H(x)\}^{11}=1+11x\{G(x)H(x)\}^6.$$

Another noteworthy formula is $$H(x)G(x^{11})-x^2G(x)H(x^{11})=1.$$ Each of these formul{\ae} is the simplest of a large class.

Endnotes

^{1}Proc. London Math. Soc.,
Ser. 1, Vol. XXV, 1894, pp. 318 – 343.

^{2}Proc. Camb. Phil. Soc., Vol. XIX, 1919,
pp. 211 – 216. A short account of the
history of the theorems is given by Mr. Hardy in a note attached to this
paper. [For Ramanujan's proofs see No. 26 of this volume.]