The generating condition for the extension of the classical Gauss series-product identity (Q2901989)

In a previous paper, [J. Algebra Appl. 9, No. 1, 123--133 (2010; Zbl 1211.17022)], the author obtained two infinite families of series-product identities based on the the classical Gauss identity NEWLINE\[NEWLINE \frac{\varphi(q^2)^2}{\varphi(q)}=\sum_{n\in \mathbb Z}q^{2n^2+n} NEWLINE\]NEWLINE by considering the characters of the fundamental modules for the affine algebra \(\widehat{\mathfrak{sl}}_k\) from two different point of views. In one case he considered \(L(\Lambda_{3m})\) and the partitions \(\{1,4m-1\}\) in the other \(L(\Lambda_{4m-1})\) and the partitions \(\{m,3m\}\).NEWLINENEWLINEIn the present paper, he gives a condition on the triple \(\{n_1,n_2,\Lambda_k\}\) so that, again, one can obtain two different interpretations of the character of the \(\widehat{\mathfrak{sl}}_k\)-module \(L(\Lambda_k)\) thus getting a family of series-product identities extending Gauss classical identity. In this fashion, he gets infinitely many new examples of such identities.