Recently, Mc Laughlin proved some results on vanishing coefficients in the series expansions of certain infinite q-products for arithmetic progressions modulo 5, modulo 7 and modulo 11 by grouping the results into several families. In this paper, we prove some new results on vanishing coefficients for arithmetic progressions modulo 7, which are not listed by Mc Laughlin. For example, we prove that if \(t \in \{1,2,3\}\) and the sequence \(\{A_n\}\) is defined by \(\sum _{n=0}^{\infty }A_nq^n := (-q^t,-q^{7-t};q^7)_{\infty }(q^{7-2t},q^{7+2t};q^{14})^3_{\infty },\) then \(A_{7n+4t}=A_{7n+6t^2+4t}=0\) for all n. Also, we prove four families of results with negative signs for arithmetic progressions modulo 7 classified by Mc Laughlin.

最近，Mc Laughlin通过将结果分成几类，证明了对等差数列模 5、模 7 和模 11的某些无穷q积的级数展开式中消失系数的一些结果。在本文中，我们证明了Mc Laughlin未列出的模7等差数列消失系数的一些新结果。例如，我们证明如果\(t \in \{1,2,3\}\)和序列\(\{A_n\}\)定义为\(\sum _{n=0}^{ \infty }A_nq^n := (-q^t,-q^{7-t};q^7)_{\infty }(q^{7-2t},q^{7+2t};q ^{14})^3_{\infty },\)然后\(A_{7n+4t}=A_{7n+6t^2+4t}=0\)对于所有n. 此外，我们证明了由 Mc Laughlin 分类的算术级数模 7 的四个带负号的结果族。