Here is the author’s introduction: “Recently B. C. Berndt, S. Bhargava and F. Garvan [Trans. Am. Math. Soc. 347, 4163-4244 (1995; Zbl 0843.33012)] provided the first proof to an identity of Ramanujan. Their proof, which is based on various modular identities, is quite difficult and complicated. In this paper, we give a much simpler proof of this identity by converting it into an identity involving the classical elliptic functions and establishing the identity by comparing their Laurent series expansions at a pole.”
And here is the identity of the title: Theorem. For , let , and ; and define
The author’s elegant proof employs the following three identities:
to show that the theorem is equivalent to:
To prove this identity, it is sufficient to show that the coefficients of the Laurent series expansions corresponding to the terms , and 3, are equal on both sides, since each side is an elliptic function with the same periods and value at 0.