The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions

Source abstract

Most of the traditional third-order mock theta functions occur naturally as specialized values of partition-rank generating functions. The same finite rank kernels also occur in the ordinary and odd unimodal-rank series studied previously in the literature. We organize a parallel collection of corner series by retaining these kernels and replacing their quadratic core weights by linear peak weights. Their root-of-unity specializations are governed by partial and false theta identities recorded in Ramanujan's Lost Notebook; in one exceptional case the corner companion is itself a mock theta function. The classical rank series and Ramanujan identities are used as inputs. Our new results include exact two- and three-color partition identities with sparse theta discrepancies and an oriented three-dimensional corner with a distinct tower. In both color cases, positive decompositions separate the colored companion into copies of the relevant odd-corner series and a corresponding decorated even-corner series. Thus both sparse discrepancies become signed differences between natural corner families. Burson's boxed modular involution provides a common rank-refined model for the two corrections, whose surviving fixed points produce the exceptional quadratic sequences. For the side rank of the three-dimensional corners we obtain an exact formula at $z=-1$, a strict parity bias, quartic and cubic root-of-unity rank-difference identities, and exact equidistribution modulo $3$ on the progression $6n+5$. The cubic identity also gives exact congruence formulas on the progressions $6n+2$ and $6n+3$. We further classify the associated binary triangular support modulo every integer and derive general congruence families.