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.

References

Can one impose conditions on the $r(n)$ so that the $a_n$ are periodic with respect to a fixed modulus, as in the product side of Rogers--Ramanujan type identities?

— On the $q$-factorization of power series  (2501.18744 - Schneider et al., 30 Jan 2025) in Section "Open questions", item 2

The following conjecture generalizes Conjecture~\ref{conj:rr-type}. For coprime $a<b$, and $0\leq r\leq (a-1)(b-1)/2$, we have $N_{a,b}{(r)}(q)=P{(r)}{a,b}(q):=\prod{i=1}\infty (1-qi){-p_{a,b}{(r)}(i)}$ for some nonnegative $(a+b)$-periodic function $p_{a,b}{(r)}$ which makes $P{(r)}_{a,b}(q)$ a normalized $W_a(a,a+b)$-character.

— Quot scheme of points on torus knot singularities  (2608.16086 - Huang et al., 17 Aug 2026) in Conjecture~\ref{conj:RR_general}, Section 1, Introduction