Extension of the q-product theorem to negative exponents

Determine whether Theorem 1, or an appropriate generalization of it, remains valid when some exponents a_n are negative, using the definition (a;q)_{-n}:=(aq^{-n};q)_n^{-1} for q-Pochhammer symbols with negative indices.

Background

Theorem 1 establishes the q-product generating formula for r_q(n) under the assumption that all a_n are nonnegative, so that the relevant q-Pochhammer symbols have nonnegative indices. The authors ask whether the result can be extended to negative exponents by adopting the standard reciprocal definition for q-Pochhammer symbols with negative indices.