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Article

Powers of Jacobi triple product, Cohen’s numbers and the Ramanujan -function

European Journal of Mathematics. 2018. Vol. 4. No. 2. P. 561-584.
Gritsenko V., Wang H.

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.