The two lost notes on page 54 of Ramanujan's lost notebook
Ramanujan wrote “see note” twice on page 54 and both notes are lost. Both missing identities are special cases of his cubic circular summation on the same page, with the Borweins’ cubic theta function a(q).
- \(q\,(A^3+B^3) = f^3(-q^6,-q^9) - f(-q,-q^4)\,a(q^5)\)
- \(a(q) = \textstyle\sum_{m,n} q^{m^2+mn+n^2}\)
- \(\text{all 12 identities verified to } 2000 \text{ coefficients}\)
Abstract
On page 54 of his lost notebook Ramanujan states two families of cubic theta-function identities and, in each family, writes "see note" next to one further claim (A^3 - B^3 in the first family, A^3 + B^3 in the second). Both notes are lost. Version 2 shows that both missing identities are special cases of another claim on the same page, Ramanujan's cubic circular summation (Entry 8.2.3 in Andrews-Berndt), and that with the Borweins' cubic theta function a(q) they read A^3 - B^3 = q^2 f^3(-q^3,-q^12) + f(-q^2,-q^3) a(q^5) and q(A^3 + B^3) = f^3(-q^6,-q^9) - f(-q,-q^4) a(q^5). The paper also gives an elementary second form, and computational evidence that the two missing combinations have no one- or two-term theta-product representation in a large explicit search space, which explains why a note was needed (a(q^5) is a binary theta series, not a product). All 12 identities are certified to 2000 coefficients by one integer-arithmetic script and independently at 50-digit precision.
Cite
Otakhon U. Kenjaev (2026). The two lost notes on page 54 of Ramanujan's lost notebook. Zenodo. https://doi.org/10.5281/zenodo.23032385