On the vanishing coefficients of odd powers of Ramanujan's theta functions (Q6612384)

In 2019, \textit{M. D. Hirschhorn} [Ramanujan J. 49, No. 2, 451--463 (2019; Zbl 1460.11059)] considered vanishing coefficients of the following two \(q\)-series expansions:\N\begin{align*}\N(-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3 &=\sum_{n=0}^\infty a(n)q^n,\tag{1}\\\N(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3 &=\sum_{n=0}^\infty b(n)q^n,\tag{2}\N\end{align*}\Nwhere\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*}\NHe proved that for any \(n\geq0\),\N\[\Na(5n+2)=a(5n+4)=b(5n+1)=b(5n+4)=0.\N\]\NSoon after, many scholars have been subsequently considered vanishing coefficients in some variants of \((1)\) and \((2)\) and established many comparable results. Very recently, \textit{D. D. Somashekara} and \textit{M. B. Thulasi} [Ramanujan J. 60, No. 2, 355--369 (2023; Zbl 1515.11046)] considered the following \(q\)-series expansion:\N\[\N(q,q^2;q^3)_\infty^3(-q^3,-q^3;q^6)_\infty=\sum_{n=0}^\infty t(n)q^n,\N\]\Nand proved that for any \(n\geq0\),\N\[\Nt(3n+2)=0.\tag{3}\N\]\NActually, \((3)\) is equivalent to\N\[\Nt'(3n+2)=0,\N\]\Nwhere the sequence \(\{t'(n)\}\) is defined by\N\[\N(q,q^2;q^3)_\infty^3=\sum_{n=0}^\infty t'(n)q^n.\N\]\N\NIn the paper under review, the author consider vanishing coefficients in the odd powers of Ramanujan's theta functions and prove a general coefficient-vanishing result. Specifically, let\N\[\Nf(-q^m,-q^{n-m})^n=(q^m,q^{n-m},q^n;q^n)^n=\sum_{k=-\infty}^\infty a_{n,m}(k)q^k,\N\]\Nwhere \(n\geq3\) is a positive odd integer and \(m\) is an integer such that \(\gcd(n,m)=1\). The author prove that for any integer \(t\),\N\[\Na_{n,m}(nt-m^2)=0.\tag{4}\N\]\NThe main ingredients in the proof of \((4)\) are two identities concerning Ramanujan's theta functions \(f(a,b)\) and certain linear transformation.\N\NInterestingly, the reviewer [Rocky Mt. J. Math. 54, No. 1, 261--267 (2024;Zbl 07823201)] also established an equivalent version of \((4)\). Both two proofs rely on two same identities. However, the approaches to the two identities are essentially different.