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Theta-function identities and the explicit formulas for theta-function and their applications. (English) Zbl 1060.11027
Let $$h_{k,n}= \frac{\varphi(e^{-\pi\sqrt{n/k}})} {k^{1/4}\varphi(e^{-\pi\sqrt{nk}})}$$ and $$h_{k,n}'= \frac{\varphi(-e^{-2\pi\sqrt{n/k}})} {k^{1/4}\varphi(-e^{-2\pi\sqrt{nk}})},$$ where $$\varphi(q)=\sum_{j=-\infty}^\infty q^{j^2}$$. Properties of $$h_{k,n}$$ and $$h_{k,n}'$$ are studied, and $$h_{k,n}$$ and $$h_{k,n}'$$ are explicitly evaluated for several values of $$k$$ and $$n$$. These evaluations come from so-called P-Q modular equations, some of which were given by Ramanujan and others are established using results in his notebooks. The paper ends with explicit values for $$\varphi(e^{-k\pi})$$ and $$\varphi(-e^{-2k\pi})$$ for $$1\leq k \leq 6$$.

##### MSC:
 11F27 Theta series; Weil representation; theta correspondences 11F20 Dedekind eta function, Dedekind sums 33E05 Elliptic functions and integrals
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##### References:
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