Analogies of Jacobi's formula

By considering Schwarz’s map for the hypergeometric differential equation with parameters $(a,b,c)=(1\mathrm{/}6,1\mathrm{/}2,1)$ or $(1\mathrm{/}12,5\mathrm{/}12,1)$, 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 $F(a,b,c;z)$.