Asymptotic expansion for the Fourier coefficients associated with the inverse of the modular discriminant function $\Delta$

Abstract

There have been a plethora of investigations carried out in studying inequalities for the Fourier coefficients of weakly holomorphic modular forms, for example, on the partition function. Recently, Bringmann, Kane, Rolen, and Tripp studied asymptotics for the $k$-colored partition function and more generally, for the fractional partitions arising from the Nekrasov-Okounkov formula which in turn allowed them to prove generalized multiplicative inequalities. Motivated by their idea to find interesting inequalities for the $k$-colored partition functions, denoted by $p_k(n)$, in this paper, we prove a family of inequalities for the $p_{24}(n)$. The main aim of this paper is to study the asymptotic expansion with an effective estimate for the error bound regarding the Fourier coefficients of the modular form $1/\Delta$ (up to a constant $c$ and a power of $q$), where $\Delta$ is the modular discriminant function and a well-known combinatorial interpretation for the associated coefficient sequence is called $24$-colored partitions, denoted by $p_{24}(n)$. Consequently, we show that $p_{24}(n)$ satisfies $2$-$\log$-concavity, Tur\'{a}n inequality of order $3$, and Laguerre inequalities of order $m$ with $2\le m\le 8$ eventually. Our method of estimations for the error term of the asymptotic expansion for $p_{24}(n)$ can be adapted in a more general paradigm where the Fourier coefficients of a certain class of Dedekind-eta quotients which are essentially a modular form of negative weight, admit a Rademacher type exact formula involving the $I$-Bessel function of positive order.