Algebraic relations between certain infinite products

Algebraic relations between certain infinite products
by Srinivasa Ramanujan

Published in the Proceedings of the London Mathematical Society, vol. XVIII, no. 2, records for 13 March 1919.

The copyright status of the paper referred to as "No. 26" below will be investigated soon.

Regarding the second footnote:

• The note by Rogers will be added to Wikisource soon.
• The note by Hardy is still under copyright.

The paper contains a small typographical error: in the second equation for ${\displaystyle \scriptstyle {G(x)}}$ the first progression operator should be in the denominator.

It was proved by Prof. L. J. Rogers[1] that

{\displaystyle {\begin{aligned}\scriptstyle {G(x)}&\scriptstyle {=1+{\frac {1}{1-x}}+{\frac {x^{4}}{(1-x)(1-x^{2})}}+{\frac {x^{9}}{(1-x)(1-x^{2})(1-x^{3})}}+\ldots }\\&\scriptstyle {={\frac {1}{(1-x)(1-x^{6})(1-x^{11})}}\ldots \times {\frac {1}{(1-x^{4})(1-x^{9})(1-x^{14})\ldots }},}\end{aligned}}}

and

{\displaystyle {\begin{aligned}\scriptstyle {H(x)}&\scriptstyle {=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})}}+\ldots }\\&\scriptstyle {={\frac {1}{(1-x^{2})(1-x^{7})(1-x^{12})\ldots }}\times {\frac {1}{(1-x^{3})(1-x^{8})(1-x^{13})\ldots }}.}\end{aligned}}}

Simpler proofs were afterwards found by Prof. Rogers and myself[2].

I have now found an algebraic relation between ${\displaystyle \scriptstyle {G(x)}}$ and ${\displaystyle \scriptstyle {H(x)}}$, viz.:

${\displaystyle \scriptstyle {H(x)\{G(x)\}^{11}-x^{2}G(x)\{H(x)\}^{11}=1+11x\{G(x)H(x)\}^{6}}}$.

Another noteworthy formula is

${\displaystyle \scriptstyle {H(x)G(x^{11})-x^{2}G(x)H(x^{11})=1}}$.

Each of these formulæ is the simplest of a large class.

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: those of Rogers, and the note by Hardy referred to, are reproduced in the notes on No. 26 in the Appendix.]

This work was published before January 1, 1926, and is in the public domain worldwide because the author died at least 100 years ago.