Modularity and the distinct rank function
Let \(\omega\) be a root of unity and \[ G(\omega,q) = \sum_{n=0}^\infty \sum_{m=0}^\infty Q(m,n) \omega^mq^n = \sum_{n=0}^\infty \frac{q^{(n^2+n)/2}}{(\omega q;q)_n}. \] The function \(Q(m,n)\) counts the number of partitions of \(n\) into distinct parts with rank \(m\). Let \[ \widehat{G}(\omega,q) = G(\omega, q^{-1}) \] and \[ \widehat{D}(\omega,q) = (1+\omega^{-1})\widehat{G}(\omega, q) + (1-\omega^{-2})\left(\psi^2(-\omega^{2},q)-1\right), \] where \[ \psi(\omega,q) = \sum_{n=0}^\infty \frac{q^{n^2}\omega^n}{(q;q^2)_n}. \] In this article, the author used the theory of \(q\)-series developed by \textit{N. J. Fine} [Basic hypergeometric series and applications, Math. Surveys and Monographs, vol. 27, Amer. Math. Soc., Providence (1988; Zbl 0647.05004)] and proved that it \(-\omega^{-1}\neq 1\) is a primitive \(m\)-th root of unity, then \[ q^{-1/12}\widehat{D}(\omega,q)\widehat{D}(\omega^{-1},q) =\frac{\eta^4(q^2)\eta^2(\omega^2,q)}{\eta^2(q)\eta^3(\omega^2,q^2)}, \] where \(\eta(q)\) is the Dedekind \(\eta\) function and \[ \eta(\omega, q) =q^{1/12}\prod_{n=1}^\infty (1-\omega q^n)(1-\omega^{-1} q^n). \] This main identity shows that a certain combination of \(\widehat{G}(\omega,q)\), \(\widehat{G}(\omega^{-1},q)\), \(\psi(-\omega^2, q)\) and \(\psi(-\omega^{-2},q)\) yields a modular form of weight 1. In the special case when \(\omega=i\), the author showed using her main result that \(q^{-1/24}\widehat{D}(i,q)\) is a modular form of weight \(1/2.\) The article is well written. This article also illustrates the fact that the proof of modularity of functions such as \(\widehat{G}(\omega,q)\) is often obtained from the study of basic hypergeometric series instead of the theory of modular forms.
scientific article; zbMATH DE number 5837843