Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Liuquan Wang (1)

1 Department of Mathematics [Wuhan]

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

Mots clés: Partitions; truncated series; Jacobi triple product; Euler's pentagonal number theorem; nonnegative coefficients

Domaines: Mathématiques [math]

Liuquan Wang. Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product. Hardy-Ramanujan Journal, 2022, Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, pp.51 -- 60. ⟨10.46298/hrj.2022.8931⟩. ⟨hal-03498175⟩