Characters and \(q\)-series in \(\mathbb Q(\sqrt 2)\)
The functions \[ R(q)=1+\sum_{n=1}^\infty \frac{q^{n(n+1)/2}}{(1+q)(1+q^2)\cdots (1+q^n)} \] and \[ L(q)=2\sum_{n=1}^\infty \frac{(-1)^nq^{n ^2}}{(1-q)(1-q^3)\cdots (1-q^{2n-1})} \] were shown by G. E. Andrews, F. Dyson, D. Hickerson and H. Cohen to be related to the field \(\mathbb Q(\sqrt{6})\). In this interesting article, the authors establish analogous results for the functions \[ W_1(q)=\sum_{n\geq 0}\frac{(q)_n(-1)^nq^{n(n+1)/2}}{(-q)_n} \] and \[ W_2(q)=\sum_{n\geq 1}\frac{(-1;q^2)_n(-q)^n}{ (q;q^2)_n}, \] where \((a;q)_n= (1-a)(1-aq)\cdots (1-aq^{n-1}).\) They show that \(W_1(q)\) and \(W_2(q)\) are associated with \(\mathbb Q(\sqrt{2})\) and prove several identities satisfied by these functions. In particular, they discover that \[ qW_1(q^8)+\frac{1}{q} W_2(q^8)=\sum_{n\geq 0} a(n)q^n, \] where \(a_n\) satisfies \[ \sum_{n\geq 1}\frac{a_n}{n^s} =\sum_{\mathfrak a\subset O_K}\frac{\chi(\mathfrak a)}{N(\mathfrak a)^s}, \] with \[ \chi(\mathfrak a)=\begin{cases} 1 & N(\mathfrak a)\equiv \pm 1\pmod{16},\\ -1 & N(\mathfrak a)\equiv \pm 7 \pmod{16},\\ 0 & \text{otherwise.}\end{cases} \]
scientific article; zbMATH DE number 2091719