Vanishing Coefficients of q^{5n+r} and q^{7n+r} in Certain Infinite q-series Expansions

Motivated by the recent work of several authors on vanishing coefficients of the arithmetic progression in certain $q$-series expansion, we study some variants of these $q$-series and prove some comparable results. For instance, if ${\sum }_{n=0}^{\mathrm{\infty }}{c}_1(n)q^n={(\pm q^2,\pm q^3;q^5)}_{\mathrm{\infty }}^2{(q,q^{14};q^{15})}_{\mathrm{\infty }}$, then $c_1(5n+3)=0$.