Analogues of Jacobi's derivative formula

TL;DR

This paper derives new formulas similar to Jacobi's derivative formula using theta constants with rational characteristics, connecting them to number representation formulas involving sums of squares.

Contribution

It introduces analogues of Jacobi's derivative formula expressed through theta constants with rational characteristics, linking to classical number theory results.

Findings

Derived analogues of Jacobi's derivative formula

Connected theta constants to sums of squares representations

Provided explicit formulas involving rational characteristics

Abstract

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.