Two generalizations of Jacobi's derivative formula

利用 Jacobi-Serre 导数的伴随构造 Jacobi 形式

Construction of Jacobi forms using adjoint of Jacobi-Serre derivative

本文研究了弱 Jacobi 形式空间上的 Oberdieck 导数，证明了其保持 Jacobi 形式性质，并推导出其与 Jacobi-Serre 导数关于 Petersson 标量积的伴随，进而揭示了 Jacobi 形式 Fourier 系数间的关系。

A Serre derivative for even weight Jacobi Forms

Using deformed or twisted Eisenstein Series, we construct a Jacobi-Serre derivative on even-weight Jacobi forms that generalizes the classical Serre derivative on modular forms. As an application, we obtain Ramanujan equations for the index $1$ Eisenstein series $E_{4,1}, E_{6,1}$ and a newly defined $E_{2,1}$. Finally, we relate the deformed Eisenstein Series directly to the classical first Jacobi theta function.

Analogues of Jacobi's derivative formula

In this paper, we obtain analogues of Jacobi's derivative formula in terms of the theta constants with rational characteristics. For this purpose, we use the arithmetic formulas of the number of representations of a natural number $n,\,\,(n=1,2,\ldots)$ as the sum of two squares, or the sum of a square and twice a square.

Formulas for Jacobi forms and generalized Frobenius partitions

Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting combinatorial structure and led to proofs of a number of congruences. In this paper, we show how Andrews' generating functions can be cast in the framework of Eichler and Zagier's Jacobi forms. This reformulation allows us to compute explicit formulas for the generalized Frobenius partition generating functions (and in fact provides formulas for further functions…

Memorandum on Dimension Formulas for Spaces of Jacobi Forms

We state ready to compute dimension formulas for the spaces of Jacobi cusp forms of integral weight $k$ and integral scalar index $m$ on subgroups of $\SL$.

An addition formula for the Jacobian theta function with applications

Liu established an addition formula for the Jacobian theta function by using the theory of elliptic functions. From this addition formula he obtained the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities. In this paper we will give an addition formula for the Jacobian theta function which is equivalent to Liu's addition formula. Based on this formula we deduce some known theta function identities as well as new identities. From these identities we shall establish certain new serie…