New identities involving sums of the tails related to real quadratic fields (Q618845)

\textit{G. Andrews, J. Jimenez-Urroz} and \textit{K. Ono} [Duke Math. J. 108, No. 3, 395--419 (2001; Zbl 1005.11048)], have obtained a number of related identities of the form \(\sum_{n\geq 0}(F(q)-F_n(q))=F(q)D(q)+E(q)\), where \(F\) is a modular infinite product, \(F_n\to F\) as \(n\to\infty\), and \(D\) is a divisor function. The coefficients of \(E\) grow much slower than those of \(FD\) and therefore, the function \(E\) may be considered as an ``error series''. In the present paper the authors construct resembling sums of tails identities which involve the functions \(f(q)=\sum_{n\geq 0}\frac{(q)_{2n}}{(-q)_{2n+1}}q^n\) and \(h(q)=\sum_{n\geq 0}\frac{(q)_{2n+1}}{(-q)_{2n+2}}q^{n+1}\) as ``error series''. The functions \(f\) and \(h\) are related to the arithmetic of \(\mathbb{Q}(\sqrt{2})\) and \(\mathbb{Q}(\sqrt{3})\). As an application, the authors also obtain formulas for the generating function of certain zeta functions for real quadratic fields at negative integers.