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Some eta-identities arising from theta series. (English) Zbl 0687.10016

In a previous paper [Abh. Math. Semin. Univ. Hamb. 58, 15-45 (1988; Zbl 0652.10020)] the author studied modular forms on the theta group which are represented by theta series with Hecke character attached to imaginary quadratic fields. Some of these double theta series split into a product of two simple series, one of which can be identified with a known function. In this way, seven identities for Dedekind’s \(\eta\)-function are obtained. Four of them appear to be new. An example is \[ \frac{\eta^ 9(2z)}{\eta^ 3(z)\eta^ 3(4z)}=\sum^{\infty}_{n=1}(\frac{-2}{n})n\quad \exp (2\pi in^ 2z/8), \] which is, like Jacobi’s identity for \(\eta^ 3\), a simple theta series representing a modular form of weight 3/2.
Reviewer: G.Köhler

MSC:

11F20 Dedekind eta function, Dedekind sums
11F27 Theta series; Weil representation; theta correspondences
11F11 Holomorphic modular forms of integral weight

Citations:

Zbl 0652.10020
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