On some new infinite q -products and their vanishing coefficients

In this paper, we prove several vanishing coefficients in arithmetic progressions modulo Formula: see text, Formula: see text and Formula: see text by using elementary Formula: see text-series manipulations and the Jacobi triple product identity for a new class of infinite Formula: see text-products, namely Formula: see text where Formula: see text is any positive integer, Formula: see text is prime and Formula: see text, Formula: see text and Formula: see text are integers that are relatively prime to Formula: see text. For instance, we prove that for any positive integer Formula: see text, if Formula: see text where Formula: see text, then Formula: see text for all non-negative integers Formula: see text.