On the infinite products derived from theta series. I.(English)Zbl 1128.11037

Let $$k$$ be an imaginary quadratic field, $$\mathbb{H}$$ the upper half-plane, and let $$q := e^{\pi i \tau}$$ with $$\tau \in \mathbb{H} \cap k$$. The authors discuss the algebraicity of certain $$\theta$$ and $$\eta$$ functions at $$q$$. For example, they note that the $$\eta$$ function is transcendental while quotients of the classical $$\theta$$ functions $$\theta_i/\theta_j$$ $$(i,j = 2,3,4)$$ are algebraic.
The bulk of the paper is devoted to L.J. Slater’s list of $$130$$ identities of the Ramanujan type. The authors present $$24$$ triples $$(a,n,t)$$ such that
$q^a\prod_{m=1}^{\infty}(1-q^{nm-t})(1-q^{nm-(n-t)})$
is algebraic, and a further $$33$$ triples such that
$q^a\prod_{m=1}^{\infty}(1+q^{nm-t})(1+q^{nm-(n-t)})$
is algebraic. They then write each of Slater’s $$130$$ identities explicitly and record whether one obtains an algebraic number, algebraic integer, or transcendental number. (Only a few cases are transcendental.)

MSC:

 11J89 Transcendence theory of elliptic and abelian functions 11F11 Holomorphic modular forms of integral weight 11F27 Theta series; Weil representation; theta correspondences 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$

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