Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions

作者：Liu, Zhi-Guo; Zhou, Nian Hong^*

来源：RAMANUJAN JOURNAL, 2022, 57(3): 1085-1123.

DOI：10.1007/s11139-021-00409-8

• Theta functions
• Fourier coefficients
• Uniform asymptotics
• Partitions
• Crank
• Rank

In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and investigated by Bringmann, Manschot, and Dousse. As applications, we establish the asymptotic monotonicity properties for the rank and crank of the integer partitions introduced and investigated by Dyson, Andrews, and Garvan.