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arXiv · 2203.07617

Analogies of Jacobi's formula

Keiji Matsumoto
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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 $F(a,b,c;z)$.

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Keiji Matsumoto. 2022-03-15. Analogies of Jacobi's formula. https://arxiv.org/abs/2203.07617

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