A note on Some Partitions Related to Ternary Quadratic Forms
Alexander E. Patkowski
TL;DR
This work connects partitions to ternary quadratic forms via a Bailey-pair framework, introducing partition counts tied to a primitive ternary form $Q(x_1,x_2,x_3)$ and analyzing their generating functions through $q$-series. The approach leverages Bailey's lemma and conjugate Bailey pairs to produce explicit generating functions and to derive structural recurrences for partition counts $B(n)$ and $\bar{B}(n)$, showing vanishing or signed recurrences outside explicit ternary-form representations $Q_1$, $Q_2$, and $Q_3$ with a growth bound of $\ll n^{3/4}$. The results illuminate how ternary forms control partition statistics and demonstrate the utility of Bailey-pair techniques in higher-dimensional form contexts. By bounding coefficients of the associated theta function $T(z)=\sum q^{Q(x_1,x_2,x_3)}$ and tying them to modular-form components, the paper links representation counts to partition theory in a precise, analyzable way. Overall, the work provides new $q$-series identities and asymptotic insights that bridge ternary quadratic forms and partition theory using conjugate Bailey-pair methods.
Abstract
We offer some partition functions related to ternary quadratic forms, and note on their upper bounds and related properties. We offer these results as an application of a simple method related to conjugate Bailey pairs presented in a prior study, further illustrating its utility.