Powers of Jacobi triple product, Cohen's numbers and the Ramanujan $\Delta$-function

TL;DR

This paper explores the powers of the Jacobi triple product, linking them to Eisenstein series, Cohen's numbers, and Ramanujan's Delta-function, providing explicit formulas and new representations for various modular forms and figurate number sums.

Contribution

It establishes that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series and derives explicit formulas for related Fourier coefficients and figurate number representations.

Findings

Eighth power of Jacobi triple product is a Jacobi--Eisenstein series of weight 4 and index 4.

Explicit formulas for Fourier coefficients of Ramanujan's Delta-function and eta powers.

New formulas for counting representations of integers as sums of higher figurate numbers.

Abstract

We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function $\Delta(\tau)=\eta^{24}(\tau)$, $\eta^{12}(\tau)$ and $\eta^{8}(\tau)$ in terms of Cohen's numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the Jacobi theta-series using the basic Jacobi forms.