The two lost notes on page 54 of Ramanujan's lost notebook

On page 54 Ramanujan wrote "see note" after A³ − B³ (Entry 8.2.1) and after A³ + B³ (Entry 8.2.2); both notes are lost. Both missing identities are special cases of his cubic circular summation (Entry 8.2.3) on the same page, with the Borweins' cubic theta function a(q) = Σ_m,n q^m²+mn+n². Everything below is computed in your browser. self-test …

1. The two missing identities, coefficient by coefficient

Th. 1 A = f(−q⁷,−q⁸), B = q f(−q²,−q¹³): A³ − B³ = q² f³(−q³,−q¹²) + f(−q²,−q³) a(q⁵)

Th. 2 A = f(−q⁴,−q¹¹), B = q f(−q,−q¹⁴): q(A³ + B³) = f³(−q⁶,−q⁹) − f(−q,−q⁴) a(q⁵)

number of coefficients N (10–3000)

Exact integer arithmetic on power series, f(a,b) = Σ a^n(n+1)/2b^n(n−1)/2 summed directly; no division, no floating point. Sections 2–3 use floating point, with f(a,b) from the Jacobi triple product (−a;ab)_∞(−b;ab)_∞(ab;ab)_∞ and a(q) from its double sum.

2. Evaluate both sides at a number q

q (−0.95 … 0.95)

3. Ramanujan's circular summation (Entry 8.2.3) for any a, b

f³(ab²,a²b) + a f³(b,a³b²) + b f³(a,a²b³) = f(a,b) · a(ab), |ab| < 1

The presets use the q of section 2. In the paper the brace of Entry 8.2.3 is rewritten as a(q) through a³ = b³ + c³ (Borweins).

4. Why a note was needed: the product obstruction

Every integer series 1 + … is uniquely ∏(1 − qⁿ)^−cₙ. It is a (generalized) eta-quotient, i.e. a finite product of theta functions, exactly when (cₙ) is periodic. The recorded cube identities have a periodic quadratic cofactor; the two lost ones do not.

n ≤ (60–600)

Paper, programs and citation: doi:10.5281/zenodo.23030496 · source code · paper page · Otakhon U. Kenjaev, 2026.