Let m,k,R,S be positive integers with 1≤ S<R/2. Consider the formal power series frac(-1)^k-1(q^S,q^R-S,q^R;q^R)_∞ ∑_0≤ n<k(-1)^nq^binomn+12R-nS(1-q^(2n+1)S). Andrews–Merca's conjecture. The coefficient of q^m in this series is non-negative. This conjecture on averaged truncations of Jacobi's triple product identity was proved independently by Mao and Yee in 2015, with later combinatorial confirmation by He, Ji, and Zang and further proofs through different q-series identities.

Andrews–Merca averaged truncation conjecture for Jacobi's triple product

Andrews and Merca conjectured that the averaged truncation has nonnegative coefficients; Guo and Zeng posed the same question independently. Mao and Yee proved the conjecture independently in 2015, and subsequent work supplied further proofs.

Known results

Mao, 2015: algebraic proof of the conjecture.
Yee, 2015: independent combinatorial proof.
He, Ji, and Zang: later combinatorial confirmation.
Wang and Yee: an averaged-truncation identity whose specialization reproves the result.

2026 stronger positivity claims

A 2026 paper claims a full combinatorial proof of a stronger normalized-tail positivity statement, implying Merca’s stronger conjecture. Another 2026 paper discusses related Jacobi-theta-tail positivity. These developments report no counterexample, gap, or retraction.

Current status (as of September 2026): The exact averaged-truncation conjecture is settled by independent 2015 proofs and later confirmations; stronger related positivity claims are reported but are unverified here.

Andrews–Merca averaged truncation conjecture for Jacobi's triple product

Primary source

Progress summary

Solutions 0