# Asymptotics for the Taylor coefficients of certain infinite products

## TL;DR

This paper investigates the asymptotic behavior of Taylor coefficients of specific infinite products involving multiple sequences of integers, extending understanding of their growth and distribution.

## Contribution

It provides new asymptotic formulas for the Taylor coefficients of complex infinite products with multiple parameters, advancing the analysis of such functions.

## Key findings

- Derived explicit asymptotic formulas for the coefficients
- Extended previous results to more general infinite products
- Enhanced understanding of the growth rates of these coefficients

## 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 $$\prod_{j=1}^J\Bigg(\prod_{k\ge 1}\big(1-q^{r_j+m_j(k-1)}\big)\big(1-q^{-r_j+m_jk}\big)\Bigg)^{\delta_j}.$$