An analogy of Jacobi's formula and its applications

We give an analogy of Jacobi’s formula, which relates the hypergeometric function with parameters $(1\mathrm{/}4,1\mathrm{/}4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.