Bilateral truncated Jacobi's identity

Recently, Andrews and Merca considered the truncated version of Euler’s pentagonal number theorem and obtained a non-negative result on the coefficients of this truncated series. Guo and Zeng showed the coefficients of two truncated Gauss’ identities are non-negative and they conjectured that the truncated Jacobi’s identity also has non-negative coefficients. Mao provided a proof of this conjecture by using an algebraic method. In this paper, we consider the bilateral truncated Jacobi’s identity and show that when the upper and lower bounds of the summation satisfy some certain restrictions, then this bilateral truncated identity has non-negative coefficients. As a corollary, we show the conjecture of Guo and Zeng holds. Our proof is purely combinatorial and mainly based on a bijection for Jacobi’s identity.