[Paper Review] Analogues of Jacobi's derivative formula III

Kazuhide Matsuda|arXiv (Cornell University)|Jun 18, 2016

Advanced Mathematical Identities6 references4 citations

TL;DR

This paper generalizes Jacobi’s derivative formula to all rational characteristics at levels $k = 3,4,5,6$ using the residue theorem, deriving explicit rational expressions for theta derivatives in terms of theta constants. The key contribution is a systematic method applicable to higher levels ($k \geq 7$), yielding product-series identities and connections to modular forms and ODEs.

ABSTRACT

In this paper, we realize high-level versions of Jacobi's derivative formula to all the rational characteristics corresponding to level $k \,\,(k=3,4,5,6).$ For this purpose, we propose the method to obtain derivative formulas by means of the residue theorem. We believe that this method can be also applied to all the rational characteristics corresponding to level $k\ge 7.$

Motivation & Objective

• To generalize Jacobi’s derivative formula to all rational characteristics at levels $k = 3,4,5,6$, extending beyond the classical case.
• To address Mumford’s open problem on whether derivatives of theta functions with rational characteristics can be expressed as rational functions of theta constants.
• To establish connections between theta derivatives and modular forms via ordinary differential equations (ODEs), particularly Riccati and Ramanujan-type systems.
• To develop a method based on the residue theorem that can be extended to higher levels ($k \geq 7$).
• To derive product-series identities through explicit formulas for theta derivatives at rational characteristics.

Proposed method

• Applies the residue theorem to elliptic functions constructed from theta functions with rational characteristics to compute derivatives.
• Uses the fundamental property that the sum of residues of an elliptic function over its fundamental parallelogram is zero.
• Constructs meromorphic functions from theta functions with specific characteristics and analyzes their poles and residues.
• Relies on known arithmetic formulas for the number of representations of integers as sums of squares or triangular numbers.
• Combines complex analysis (residue calculus) with number-theoretic identities (e.g., $S_2(n)$, $S_{1,2}(n)$) to derive closed-form expressions.
• Employs Farkas and Kra’s theory of theta functions with rational characteristics to express derivatives as rational functions of theta constants.

Experimental results

Research questions

• RQ1 Can Jacobi’s derivative formula be generalized to all rational characteristics at levels $k = 3,4,5,6$?
• RQ2 Can the derivative of a theta function with rational characteristic be expressed as a rational function of other theta constants?
• RQ3 Does the residue theorem provide a systematic method to derive such derivative formulas across all rational characteristics at a given level?
• RQ4 What are the connections between these derivative formulas and modular forms or ODEs such as Ramanujan’s system?
• RQ5 Can the method be extended to levels $k \geq 7$?

Key findings

• The paper derives explicit formulas for $\theta'\left[\begin{array}{c}0 \\ 1/4\end{array}\right](0,\tau)$ and $\theta'\left[\begin{array}{c}0 \\ 3/4\end{array}\right](0,\tau)$ as rational functions of theta constants, specifically: $\theta'\left[\begin{array}{c}0 \\ 1/4\end{array}\right] = -\pi \theta\left[\begin{array}{c}0 \\ 1/4\end{array}\right] \theta\left[\begin{array}{c}1 \\ 0\end{array}\right](0,4\tau)\left\{\sqrt{2}\theta\left[\begin{array}{c}0 \\ 0\end{array}\right](0,2\tau) - \theta\left[\begin{array}{c}1 \\ 0\end{array}\right](0,4\tau)\right\}$.
• Similarly, $\theta'\left[\begin{array}{c}0 \\ 3/4\end{array}\right] = -\pi \theta\left[\begin{array}{c}0 \\ 3/4\end{array}\right] \theta\left[\begin{array}{c}1 \\ 0\end{array}\right](0,4\tau)\left\{\sqrt{2}\theta\left[\begin{array}{c}0 \\ 0\end{array}\right](0,2\tau) + \theta\left[\begin{array}{c}1 \\ 0\end{array}\right](0,4\tau)\right\}$.
• The difference of logarithmic derivatives yields $\frac{\theta'\left[\begin{array}{c}0 \\ 1/4\end{array}\right]}{\theta\left[\begin{array}{c}0 \\ 1/4\end{array}\right]} - \frac{\theta'\left[\begin{array}{c}0 \\ 3/4\end{array}\right]}{\theta\left[\begin{array}{c}0 \\ 3/4\end{array}\right]} = 2\pi \theta^2\left[\begin{array}{c}1 \\ 0\end{array}\right](0,4\tau)$, linking to the number of representations of odd integers as sums of two squares.
• The sum of logarithmic derivatives gives $\frac{\theta'\left[\begin{array}{c}0 \\ 1/4\end{array}\right]}{\theta\left[\begin{array}{c}0 \\ 1/4\end{array}\right]} + \frac{\theta'\left[\begin{array}{c}0 \\ 3/4\end{array}\right]}{\theta\left[\begin{array}{c}0 \\ 3/4\end{array}\right]} = -2\sqrt{2}\pi \theta\left[\begin{array}{c}1 \\ 0\end{array}\right](0,4\tau) \theta\left[\begin{array}{c}0 \\ 0\end{array}\right](0,2\tau)$, connecting to representations as sums of squares and triangular numbers.
• The method based on the residue theorem successfully generalizes Jacobi’s formula to all rational characteristics at levels $k=3,4,5,6$, and the author believes it extends to $k \geq 7$.
• The results produce numerous product-series identities involving theta constants and provide a pathway to derive ODEs satisfied by modular forms, such as Riccati equations for ratios of theta constants.