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

Kim, Daeyeoul; Koo, Ja Kyung

J. Korean Math. Soc. 44, No. 1, 55-107 (2007).

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.)

Reviewer: Jeremy Lovejoy (Paris)

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, \({}_r\phi_s\)

Keywords:

algebraic and transcendental numbers; theta series; Slater’s identities

Software:

RRtools; recpf