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± n, microscopic q-supercongruences, and cyclotomic/p-adic consequences within a single framework. For S(q,a):=∑n≥ 0(q2n+1(aq,q/a;q2)n)/((−q2;q2)n), we prove, for odd n, a finite theta evaluation at a=q± n, equivalently a microscopic q-supercongruence. We prove an analogous finite theta evaluation for the ν-type denominator Nm(q,a):=∑j=0m q2j+1(aq,q/a;q2)j/((−q;q2)j+1). We then introduce the one-parameter denominator family T(q;b):=∑n≥ 0 q2n(q;q2)n2/(q2;q2)n(b;q2)n, whose specializations include the ω-type shifted denominator (b=q3) and the two-color family (b=q2k). For its truncated a-extension we obtain an exact evaluation at a=q± n, deduce microscopic congruences modulo (a−qn)(1−aqn), and derive cyclotomic consequences, including zero congruences in the q2k-subfamily and nontrivial p-adic versions for the cases b=q and b=q3.
Related
- Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta-functions
- The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions
- Mock Modularity, Resurgence, and Dual False Theta Functions
- Resurgent rigidity of mock theta functions: uniqueness and natural boundary crossing
- On sign patterns of Fourier coefficients of theta quotients
BibTeX
@misc{bachraoui2026qmicroscope,
title = {{A $q$-microscope for mock-theta denominator patterns}},
author = {Mohamed El Bachraoui},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05961},
eprint = {2609.05961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05961},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A q-microscope for mock-theta denominator patterns
AU - Mohamed El Bachraoui
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05961
UR - https://arxiv.org/abs/2609.05961
AB - 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^± n, microscopic q-supercongruences, and cyclotomic/p-adic consequences within a single framework. For S(q,a):=∑_n≥ 0(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^± n, equivalently a microscopic q-supercongruence. We prove an analogous finite theta evaluation for the ν-type denominator N_m(q,a):=∑_j=0^m q^2j+1(aq,q/a;q^2)_j/((−q;q^2)_j+1). We then introduce the one-parameter denominator family T(q;b):=∑_n≥ 0 q^2n(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^± 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.
ER -