arXiv subjects

Matthew Ortiz

Publications and source records attributed to Matthew Ortiz.

4 recordsLinked to original sources

arXiv2026

Recursions for Mock Theta Functions

We establish weighted recursions for the coefficients of Ramanujan's third order mock theta functions $f$ and $\omega$. Specifically, we apply a holomorphic projection operator to vector-valued Rankin-Cohen brackets of completed mock theta series and their shadows. By employing a vector-valued framework, we exploit the vanishing of certain spaces of vector-valued cusp forms. Our proof is AI-assisted and prioritizes accessibility, allowing for straightforward customization and replication within the broader research community.

math.NT↗

arXiv2026

Weighted Recursions for the Smallest Parts Function

We establish new polynomial-weighted recursions for Andrews' smallest parts function. Our results use the generating series for the spt function, a harmonic Maass form of weight 3/2, paired with the Dedekind eta function via the Rankin-Cohen bracket. In weights larger than 2, we find a nontrivial quasimodular component, which we determine for the relevant weights. We apply the holomorphic projection operator and the vanishing of cusp form spaces of small enough weight to obtain our results.

math.NT↗

arXiv2026

Weighted Recursions for Hurwitz Class Numbers

We establish new recursions for Hurwitz class numbers with polynomial weights. In contrast to previous recursions, our results decouple class numbers of even and odd discriminants. Our main tool is the vector-valued holomorphic projection operator applied to mock modular forms. We invoke representation theory to connect the relevant spaces of vector-valued modular forms to spaces of classical new and old forms. We thereby leverage the vanishing of spaces of vector-valued cusp forms not available in the scalar case.

math.NT↗

arXiv2025

The higher order partial derivatives of Okamoto's function with respect to the parameter

Let $\{F_a: a\in(0,1)\}$ be Okamoto's family of continuous self-affine functions, introduced in [{\em Proc. Japan Acad. Ser. A Math. Sci.} {\bf 81} (2005), no. 3, 47--50]. This family includes well-known ``pathological" examples such as Cantor's devil's staircase and Perkins' continuous but nowhere differentiable function. It is well known that $F_a(x)$ is real analytic in $a$ for every $x\in[0,1]$. We introduce the functions \[ M_{k,a}(x):=\frac{\partial^k}{\partial a^k}F_a(x), \qquad k\in\mathbb{N}, \quad x\in[0,1]. \] We compute the box-counting dimension of the graph of $M_{k,a}$, characterize its differentiability, and investigate in detail the set of points where $M_{k,a}$ has an infinite derivative. While some of our results are similar to the known facts about Okamoto's function, there are also some notable differences and surprising new phenomena that arise when considering the higher order partial derivatives of $F_a$.

math.CA↗