Periodicity conditions for q-factorization exponents

Determine conditions on the coefficients r(n) that guarantee the q-factorization exponents a_n are periodic modulo a prescribed fixed modulus, as occurs on the product side of Rogers–Ramanujan-type identities.

Background

The paper observes that Rogers–Ramanujan-type identities often have product sides whose exponents follow periodic patterns. Since Theorem 2 expresses each exponent a_n directly in terms of the coefficients r(n), the unresolved issue is to characterize coefficient sequences whose associated exponent sequence is periodic with respect to a fixed modulus.