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.)