Mock Functions: q-Series & Modular Phenomena

• Mock functions are specialized q-hypergeometric series, notably Ramanujan's mock theta functions, that exhibit mock modular behavior with non-holomorphic completions.
• Constructed via Bailey chains and q-hypergeometric methods, they bridge combinatorics, analytic number theory, and the theory of modular forms.
• They play a significant role in partition theory, representation theory, and quantum modular forms, influencing both pure mathematics and mathematical physics.

Mock functions, in the context of modern mathematics and mathematical physics, refer primarily to mock theta functions—distinguished q-hypergeometric series defined originally by S.~Ramanujan—as specific instances of mock modular forms. These series encapsulate a variety of phenomena at the interface of combinatorics, analytic number theory, harmonic weak Maass forms, quantum modular forms, and representation theory of affine Lie algebras and related structures. Distinguished by their "mock" modular transformation behavior, mock theta functions and their recent generalizations have become central objects across several mathematical subfields.

1. Definition and Structural Properties

Mock theta functions are q-series $F(q)$, typically defined for $|q|<1$ (e.g., via sums $\sum_{n\geq0} a(n)q^n$), that exhibit "mock" modular behavior: although they are not modular forms, their lack of modularity can be quantified and, in modern theory, "completed" via explicit non-holomorphic corrections (as in Zwegers' framework). A mock theta function is fundamentally characterized as the holomorphic part of a weight $1/2$ harmonic weak Maass form, with a non-holomorphic modular completion whose image under the $\xi$-operator is a unary theta function—the so-called "shadow."

Pure mock theta functions contrast with "mixed" mock modular forms, which involve linear combinations or products of modular and mock modular components. For pure mock theta functions, after completion, the modularity defect is entirely rectified by a single unary theta function. In contrast, Bailey-chain constructions and related iterative schemes often proliferate mixed mock modular forms unless additional manipulation—such as a careful change of base in Bailey pairs—is used to recover pure mock modularity (Lovejoy et al., 2012).

The canonical examples are Ramanujan's mock theta functions, including third, fifth, seventh, and tenth order assemblies, each of which plays a role in the greater ecosystem of q-series, partition functions, and modular phenomena.

2. Construction Mechanisms: Bailey Chains and q-Hypergeometric Formats

A primary modern mechanism for constructing mock theta functions is via the Bailey pair formalism and the Bailey chain (Lovejoy et al., 2012). Given sequences $(a_n, B_n)$ forming a Bailey pair with respect to $a$, one generates multisum q-series with specifically controlled transformation properties. The recursion,

$$B_n = \sum_{k=0}^n \frac{(q)_{n-k}(a q)_{n+k-1}}{(q)_{n-k}(a q)_{n+k}} a_k,$$

and the associated iterative "chain" formulas (see Eqs. (1.5)--(1.6) in (Lovejoy et al., 2012)), allow systematic generation of multisums expressible in terms of Appell--Lerch sums or indefinite theta series. However, without a change of base (e.g., using Bressoud-Ismail-Stanton base change as in Lemma 1.1), these multisums typically yield mixed mock modularity. Only via intricate control—altering the base and parameters as in Eqs. (1.12)--(1.13)—do we obtain pure mock theta functions directly associated with classically studied $q$-series.

Further, representation of mock theta functions as specializations or linear combinations of so-called universal mock theta functions (e.g., $R(z,q), H(z,q), K(z,q)$) has illuminated their combinatorial and analytic structure. Many classical mock theta functions can be obtained as

$f(q) = \sum_{n\ge0} (-q;q)_n^2 q^n = R(-1, q),$

where $R(z,q)$ is a universal two-variable q-series (Garvan, 2014).

3. Modular and Quantum Modular Structure

A defining property of mock theta functions is their failure to be modular forms, but with a precisely quantifiable "defect." Zwegers' theory recasts these functions by adding explicit non-holomorphic correction terms constructed analytically (e.g., using error functions and indefinite theta series), yielding the completed function that transforms as a weight $1/2$ modular form (Kac et al., 2015, Wakimoto, 2022).

Mathematically, mock theta functions $F(q)$ have the property that for many (but not all) roots of unity $\zeta$, there exist (weakly holomorphic) modular forms $M_{\zeta}(q)$ and explicit $q$-shifts such that as $q$ radially approaches $\zeta$,

$F(q) - q^a M_{\zeta}(q)$

remains bounded, but no single $M(q)$ works simultaneously for all roots of unity. This is now interpreted through their association with harmonic Maass forms and their "shadows."

The paper of radial limits quantifies the behavior of mock theta functions at the cusps of the upper half-plane and connects them with quantum modular forms—a class of functions with analytic or quantum modularity (on $\mathbb{Q}$) instead of holomorphy in the upper half-plane (Bringmann et al., 2014, Folsom et al., 4 Jul 2025). For instance, limiting constants at cusps arising from such radial limits often form genuine quantum modular forms of weight $1/2$.

4. Representation, Identities, and Universal Constructions

A prominent theme in contemporary research is uncovering multiple representations of mock theta functions—in Eulerian, Appell--Lerch, and Hecke-type double sum forms. Unified approaches using parameterized q-hypergeometric transformations recover these identities and reveal deep connections between various mock theta functions, sometimes of differing orders, via underlying two-parameter identities (Chen et al., 2018). For example,

$$m(x, q, z) = \frac{1}{j(z,q)} \sum_{n\in \mathbb{Z}} \frac{(-1)^n q^{n(n+1)/2} z^n}{1 - x q^n},$$

is a canonical building block, and rewriting mock theta functions in terms of such Appell--Lerch sums illuminates their modular and analytic features.

Recent works have produced comprehensive generalizations—not only expressing classical mock theta functions as specializations of "optimal mock Jacobi theta functions" indexed by genus-zero groups (Cheng et al., 2016), but also extending them to bilateral, parameter-rich series (see the "complete generalized new mock theta functions") and deriving continued fraction or ${}_2\phi_1$ hypergeometric expansions (Tiwari et al., 2023).

5. Combinatorics, Partition Theory, and Arithmetic Behavior

Mock theta functions occupy a central position in partition theory and combinatorial number theory. For example, the coefficients of the third-order function $\omega(q)$ generate the number of partitions where each odd part is less than twice the smallest part (Ballantine et al., 2022, Baruah et al., 2019). Universal mock theta functions serve as generating functions for refinements of partition statistics, including Dyson’s rank, overpartition rank, and spt-crank functions (Garvan, 2014).

The arithmetic properties of their coefficients have also been intensely scrutinized. Notably, for many (but not all) mock theta functions and related weakly holomorphic modular forms, there are no simple linear congruences modulo $2$ or $3$, in contrast to the ordinary partition function $p(n)$, as established using modular transformation analysis and local expansion at cusps (Ahlgren et al., 2013). Additionally, extensive work (see (Wang, 2020)) studies the parity of coefficients, with several functions exhibiting nearly all coefficients even (parity type $(1,0)$), while others display half or sparser distributions of odd coefficients.

Strong congruence and asymptotic results—mirroring classical Ramanujan congruences for $p(n)$—have been established for partition functions attached to mock theta functions. For example, congruences modulo 5, 9, and 15 for partition statistics derived from mock theta generating functions have been proven, illustrating the deep interplay between combinatorics and modular phenomena (Chern et al., 2017).

6. Applications in Representation Theory, Mathematical Physics, and Future Directions

Mock theta functions appear as building blocks for vector-valued modular forms associated with affine Lie (super)algebras, string functions in parafermionic conformal field theories, and characters of admissible highest-weight representations (Borozenets et al., 23 Sep 2024, Borozenets et al., 5 Feb 2025). In these settings, explicit "mock theta conjecture-like" identities express string functions at fractional levels in terms of Ramanujan’s mock theta functions, solidifying their analytic and algebraic role in representation theory.

Moreover, the structure admits generalizations to higher depth (greater than one) mock theta functions, with holomorphic parts whose images under modular lowering operators land in lower-depth spaces, broadening the landscape of mock modular phenomena and enabling new q-hypergeometric constructions (Males et al., 2021).

The connection to quantum modular forms and the emergence of antiquantum $q$-series identities provide refined perspectives on the asymptotic behavior at roots of unity, corroborating the original asymptotic definitions offered by Ramanujan and further intertwining the analytic and arithmetic aspects of these functions (Folsom et al., 4 Jul 2025).

A plausible implication is that continued investigation—particularly into universal constructions via q-difference equations and the structure of positive-level admissible string functions using Appell functions—will both unify and deepen the interplay of mock modularity, quantum modularity, and partition combinatorics.

7. Summary Table: Key Representational Tools for Mock Theta Functions

Method/Representation	Description	Canonical Example or Formula
Bailey chain & pairs	Iterative q-hypergeometric multisum construction	$$B_n = \sum_{k=0}^n \frac{(q)_{n-k}(a q)_{n+k-1}}{(q)_{n-k}(a q)_{n+k}} a_k$$
Appell–Lerch sum	Fundamental building block for analytic and modular properties	$$m(x, q, z) = \frac{1}{j(z,q)} \sum_{n\in\mathbb{Z}} \frac{(-1)^n q^{n(n+1)/2} z^n}{1 - x q^n}$$
Universal mock theta functions	Two-variable q-series specializing to classical mock theta functions	$$R(z,q) = \sum_{n=0}^\infty \frac{q^{n^2}}{(zq;q)_n (z^{-1}q;q)_n}$$
Indefinite theta series	Sums over lattice points with sign constraints, modular completions	$$f_{a,b,c}(x,y,q) = \sum_{\substack{r,s\in\mathbb{Z}\\operatorname{sgn}(r)=\operatorname{sgn}(s)}} \operatorname{sgn}(r) (-1)^{r+s} x^r y^s q^{ar^2 + brs + cs^2}$$
Bilateral and generalized series	Extensions and generalizations with parameter-rich bilateral series	$$J_0(t,a,z;q) = \sum_{n=0}^\infty (t;q)_n q^{2n^2 - 3n + n a} z^{2n}/(a;q)_n$$

These representational frameworks enable both a unified analytic description of existing mock theta functions and a systematic approach for constructing new ones with controlled modular, quantum modular, and arithmetic properties. Their broad applicability in combinatorics, representation theory, and quantum topology underscores the ongoing vitality and depth of the field.