On a generalization of vanishing coefficients in two \(q\)-series expansions (Q6614574)

The authors consider two sequences, namely\N\[\N\sum_{n=-\infty}^{\infty}g(n)q^{n}=(q^{2t},q^{5-2t};q^{5})_{\infty}({\pm}q^{5-t},{\pm}q^{5+t};q^{10})_{\infty}^{2},\N\]\Nand \N\[\N\sum_{n=-\infty}^{\infty}h(n)q^{n}=({\pm}q^{2t},{\pm}q^{5-2t};q^{5})_{\infty}^{2}(q^{2t},q^{10-2t};q^{10})_{\infty}.\N\]\N\NThey prove that \(g(5n+t)=h(5n+3t^{2}-2t)=0\), where \(t\ge1\), and \(5\) does not divid \(t\). Furthermore, they generalize the results obtained by \textit{M. D. Hirschhorn} [Ramanujan J. 49, No. 2, 451--463 (2019; Zbl 1460.11059)], \textit{D. D. Somashekara} and \textit{M. B. Thulasi} [Ramanujan J. 60, No. 2, 355--369 (2023; Zbl 1515.11046)], \textit{D. Tang} [Int. J. Number Theory 15, No. 4, 763--773 (2019; Zbl 1459.11117)], \textit{D. Tang} and \textit{Ernest. X. W. Xia} [Ramanujan J. 53, No. 3, 705--724 (2020; Zbl 1467.33013)], and \textit{D. Q. J. Dou} and \textit{J. Xiao} [Ramanujan J. 54, No. 3, 475--484 (2021; Zbl 1466.33012)].