Asymptotics for the Taylor coefficients of certain infinite products.

Let (m 1 , ... , m J) and (r 1 , ... , r J) be two sequences of J positive integers satisfying 1 ≤ r j < m j for all j = 1 , ... , J . Let (δ 1 , ... , δ J) be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product ∏ j = 1 J (∏ k ≥ 1 (1 - q r j + m j (k - 1) ) (1 - q - r j + m j k ) ) δ j. Our work generalizes many known results, including an asymptotic formula due to Lehner for the partition function arising from the first Rogers–Ramanujan identity. The main technique used here is based on Rademacher's circle method. [ABSTRACT FROM AUTHOR]