Analogies of Jacobi's formula

TL;DR

This paper explores analogies of Jacobi's formula using Schwarz's map for specific hypergeometric equations, deriving new functional equations for hypergeometric series related to theta functions.

Contribution

It introduces novel analogies of Jacobi's formula via Schwarz's map for particular hypergeometric equations, expanding the understanding of theta functions and hypergeometric series.

Findings

Derived functional equations for hypergeometric series

Established analogies between theta constants and hypergeometric functions

Connected Schwarz's map with classical formulas in special functions

Abstract

By considering Schwarz's map for the hypergeometric differential equation with parameters $(a,b,c)=(1/6,1/2,1)$ or $(1/12,5/12,1)$, we give some analogies of Jacobi's formula $\vartheta_{00}(\tau)^2= F(1/2,1/2,1;\lambda(\tau))$, where $\vartheta_{00}(\tau)$ and $\lambda(\tau)$ are the theta constant and the lambda function defined on the upper-half plane, and $F(a,b,c;z)$ is the hypergeometric series defined on the unit disk. As applications of our formulas, we give several functional equations for $F(a,b,c;z)$.