A $q$-microscope for mock-theta denominator patterns

Abstract

We give a systematic creative-microscoping treatment of truncated basic hypergeometric sums attached to three classical third-order mock theta denominator patterns, represented by $ϕ(q)$, $ω(q)$, and $ν(q)$. The main conceptual point is that these denominator patterns naturally admit exact finite theta evaluations at $a=q^{\pm n}$, microscopic $q$-supercongruences, and cyclotomic/$p$-adic consequences within a single framework. For \[ S(q,a):=\sum_{n\ge 0}\frac{q^{2n+1}(aq,q/a;q^2)_n}{(-q^2;q^2)_n}, \] we prove, for odd $n$, a finite theta evaluation at $a=q^{\pm n}$, equivalently a microscopic $q$-supercongruence. We prove an analogous finite theta evaluation for the $ν$-type denominator \[ N_m(q,a):=\sum_{j=0}^{m} q^{2j+1}\frac{(aq,q/a;q^2)_j}{(-q;q^2)_{j+1}}. \] We then introduce the one-parameter denominator family \[ \T(q;b):=\sum_{n\ge 0} q^{2n}\frac{(q;q^2)_n^2}{(q^2;q^2)_n(b;q^2)_n}, \] whose specializations include the $ω$-type shifted denominator ($b=q^3$) and the two-color family ($b=q^{2k}$). For its truncated $a$-extension we obtain an exact evaluation at $a=q^{\pm n}$, deduce microscopic congruences modulo $(a-q^n)(1-aq^n)$, and derive cyclotomic consequences, including zero congruences in the $q^{2k}$-subfamily and nontrivial $p$-adic versions for the cases $b=q$ and $b=q^3$.