On a generalization of 5-dissections of some infinite \(q\)-products (Q6596363)

In 2018, \textit{M. D. Hirschhorn} [Ramanujan J. 49, No. 2, 451--463 (2019; Zbl 1460.11059)] studied vanishing coefficients with arithmetic progressions in two \(q\)-series expansions. Specifically, he proved the following remarkable identities, namely, if \N\begin{align*} \N\sum_{n=0}^\infty a(n)q^n &=(-q,-q^4;q^5)_\infty (q,q^9;q^{10})_\infty^3,\tag{1}\\\N\sum_{n=0}^\infty b(n)q^n &=(-q^2,-q^3;q^5)_\infty (q^3,q^7;q^{10})_\infty^3,\tag{2} \N\end{align*} \Nthen\N\[\Na(5n+2)=a(5n+4)=b(5n+1)=b(5n+4)=0,\N\]\Nwhere we always assume that \(|q|<1\) and adopt the following notation:\N\begin{align*}\N(a;q)_\infty &:=\prod_{j=0}^\infty(1-aq^j),\\\N(a_1,a_2,\ldots,a_m;q)_\infty &:=(a_1;q)_\infty(a_2;q)_\infty\cdots (a_m;q)_\infty.\N\end{align*}\NMotivated by these results, the reviewer [Int. J. Number Theory 15, No. 4, 763--773 (2019; Zbl 1459.11117)] considered some variants of $(1)$ and $(2)$ and obtained some comparable identities. Soon after, \textit{E. X. W. Xia} and the reviewer [Ramanujan J. 53, No. 3, 705--724 (2020; Zbl 1467.33013)] obtained the \(5\)-dissection formulas for two \(q\)-series expansions, which were introduced by the reviewer. More precisely, they proved that\N\begin{multline*}\N(-q,-q^4;q^5)_\infty^2(q^4,q^6;q^{10})_\infty\\\N=\frac{1}{(q^5,q^{20};q^{25})_\infty^2(q^{10},q^{40};q^{50})_\infty} +\frac{2}{(q^5,q^{10},q^{15},q^{20};q^{25})_\infty (q^{10},q^{40};q^{50})_\infty}\\\N+\frac{1}{(q^5,q^{20};q^{25})_\infty^2(q^{20},q^{30};q^{50})_\infty} +\frac{1}{(q^{10},q^{15};q^{25})_\infty^2(q^{20},q^{30};q^{50})_\infty}.\N\end{multline*}\NThere is a similar \(5\)-dissection formula for \((-q^2,-q^3;q^5)_\infty^2(q^2,q^8;q^{10})_\infty\). Soon after, \textit{D. Q. J. Dou} and \textit{J. Xiao} [Ramanujan J. 54, No. 3, 475--484 (2021; Zbl 1466.33012)] established the \(5\)-dissection formulas for \((q,q^4;q^5)_\infty(q^3,q^7;q^{10})_\infty^2\) and \((q^2,q^3;q^5)_\infty(q,q^9;q^{10})_\infty^2\).\N\NThe main purpose of this paper is to generalize the above \(5\)-dissection formulas to an finite family case. Specifically, the authors established \(5\)-dissection formulas for the following two infinite families of \(q\)-series expansions:\N\begin{align*}\N&(\pm q^t,\pm q^{5\ell-t},q^{5\ell})_\infty^2(\mp q^t,\mp q^{10\ell-t}; q^{10\ell})_\infty,\\\N&(\pm q^t,\pm q^{5\ell-t},q^{5\ell})_\infty^2(q^{5\ell-2t},q^{5\ell+2t}; q^{10\ell})_\infty.\N\end{align*}\NWherever ambiguity signs appear, signs within the product are to be taken either both upper or both lower. Moreover, the authors also consider \(5\)-dissections in another two infinite families of \(q\)-series expansions, including \((\pm q^t,\pm q^{5\ell-t},q^{5\ell})_\infty^2(\pm q^t,\pm q^{10\ell-t}; q^{10\ell})_\infty\).