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Asymptotics for the Taylor coefficients of certain infinite products

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Abstract

Let \((m_1,\ldots ,m_J)\) and \((r_1,\ldots ,r_J)\) be two sequences of J positive integers satisfying \(1\le r_j< m_j\) for all \(j=1,\ldots ,J\). Let \((\delta _1,\ldots ,\delta _J)\) be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product

$$\begin{aligned} \prod _{j=1}^J\Bigg (\prod _{k\ge 1}\big (1-q^{r_j+m_j(k-1)}\big )\big (1-q^{-r_j+m_jk}\big )\Bigg )^{\delta _j}. \end{aligned}$$

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.

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Acknowledgements

I am grateful to the referee for careful reading and valuable comments. I also want to thank George Andrews for helpful suggestions and for pointing out a number of useful references.

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  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Shane Chern

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Chern, S. Asymptotics for the Taylor coefficients of certain infinite products. Ramanujan J 55, 987–1014 (2021). https://doi.org/10.1007/s11139-020-00273-y

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