Ramanujan's Third-Order Mock Theta Functions

• Ramanujan's third-order mock theta functions are q-hypergeometric series that serve as the holomorphic parts of weight 1/2 harmonic Maass forms with deep modular connections.
• They are unified through a universal two-variable mock theta function and admit Hecke–Rogers double-sum representations, revealing intricate transformation and asymptotic properties.
• These functions bridge analytic number theory, combinatorial partition theory, and representation theory, offering a fertile ground for advances in quantum modularity and automorphic forms.

Ramanujan’s third-order mock theta functions are a distinguished subset of his broader collection of mock theta functions, introduced in his final letter to Hardy in 1920. These functions, notable for their q-hypergeometric expansions and striking transformation properties, occupy a central position in the analytic theory of mock modular forms. Their unifying structure, deep connections to partition theory, and modularity phenomena have attracted ongoing research interest spanning analytic, combinatorial, and representation-theoretic domains.

1. Classical Definitions and Universal Structure

Ramanujan’s third-order mock theta functions are explicit $q$-hypergeometric series, convergent for $|q|<1$. The five standard functions of order three are $f(q)$, $\phi(q)$, $\psi(q)$, $\chi(q)$, and $\omega(q)$. Each can be realized as a specialization of a two-variable "universal mock theta function" $g(x, q)$: $$g(x,q) = x^{-1} \left(-1 + \frac{1}{(x; q)_\infty (x^{-1} q; q)_\infty} \right) = x^{-1} \sum_{n=0}^{\infty} \frac{q^{n^2}}{(x; q)_n (x^{-1} q; q)_n}$$ with $(a; q)_n = \prod_{k=0}^{n-1}(1 - a q^k)$, $(a; q)_\infty = \prod_{k=0}^{\infty}(1 - a q^k)$.

The classical third-order mock theta functions are obtained by setting $x$ to suitable roots of unity:

• $f(q) = g(-1, q)$
• $\phi(q) = g(i, q)$
• $\psi(q), \chi(q), \omega(q)$ by further specialization of $x$ as appropriate (Garvan, 2014).

These functions admit the following explicit $q$-hypergeometric forms: $$\begin{aligned} f(q) &= \sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q; q)_n^2} \ \phi(q) &= \sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q^2; q^2)_n} \ \psi(q) &= \sum_{n=0}^{\infty} \frac{q^{n(n+1)}}{(-q^2; q^2)_n} \ \chi(q) &= \sum_{n=0}^{\infty} (-1)^n \frac{q^{n^2}}{(-q^2; q^2)_n} \ \omega(q) &= \sum_{n=0}^{\infty} \frac{q^{2n(n+1)}}{(q; q^2)_{n+1}^2} \end{aligned}$$ The universal structure is further evident in the realization of all third-order functions as specializations of $g_3(z, q)$ (Schneider, 2016): $$g_3(z, q) = \sum_{n=1}^{\infty} \frac{q^{n(n-1)}}{(z; q)_n (z^{-1} q; q)_n}$$ with, for example, $f(q) = g_3(-1, q)$, $\phi(q) = g_3(\omega, q)$, and $\psi(q) = g_3(\omega^2, q)$ for $\omega = e^{2\pi i/3}$.

2. Hecke–Rogers Type Double-Sum Expansions

These mock theta functions admit canonical Hecke–Rogers double sum representations via their embedding into "universal" generating functions (Garvan, 2014). For instance, Garvan established that the function

$$R(z, q) = \sum_{n=0}^{\infty} \frac{q^{n^2}}{(zq; q)_n (z^{-1}q; q)_n}$$

satisfies a two-variable double-sum identity: $$(1+z) (zq; q)_\infty (z^{-1}q; q)_\infty (q)_\infty = \sum_{n=0}^{\infty} \sum_{j=0}^{\lfloor n/2 \rfloor} (-1)^{n+j} (z^{n-3j} + z^{3j-n}) q^{n^2 - 3j^2 + (n-j)}$$ Specializing $z$ to roots of unity gives double-sum expansions for $f(q)$ and the other third-order functions. For $f(q)$, this yields

$$f(q) = (q; q)_\infty (q^2; q^2)_\infty \sum_{n=0}^{\infty} \sum_{m = -\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor} \operatorname{sgn}(m)(-1)^{n+m} q^{n^2 - 3m^2 + (n-m)}$$

with analogous formulas for the other classical third-order mock thetas (Garvan, 2014, Mortenson, 2012).

The existence of these double-sum representations is rooted in the machinery of Bailey pairs, limiting forms of Bailey's lemma, and bilinear hypergeometric transformations.

3. Modular and Harmonic Maass Form Properties

Despite their nonmodular $q$-series appearance, Ramanujan's third-order mock theta functions are precisely the holomorphic parts of weight $1/2$ harmonic Maass forms, with nonholomorphic shadows given by unary theta series. For $f(q)$, Bringmann and Ono completed Andrews's asymptotic and conjectural formula and established an exact Rademacher-type formula for the Fourier coefficients $\alpha_f(n)$ (Andersen et al., 2 Sep 2025): $$\alpha_f(n) = \frac{2\pi}{(24n-1)^{1/4}} \sum_{c=1}^{\infty} \frac{1}{c} \sqrt{\frac{c}{3}} \sum_{b \pmod{6c} \atop b^2 \equiv -(24n-1) \bmod 24c} \binom{-12}{b} \sin\left( \frac{\pi b}{6c} \right) I_{1/2}\left( \frac{\pi \sqrt{24n-1}}{6c} \right)$$ where $I_{1/2}(x) = \sqrt{2/(\pi x)}\sinh(x)$, and $\binom{-12}{b}$ is a Gauss symbol arising from the Weil representation attached to a rank-1 lattice.

The completion theory, pioneered by Zwegers, expresses each mock theta as the holomorphic part of a real-analytic Maass form, with completion

$$\widehat{f}(\tau) = f(e^{2\pi i \tau}) + (\text{explicit nonholomorphic Eichler integral})$$

transforming with the correct half-integral weight, up to a shadow determined by a suitable theta series (Bringmann et al., 2011, Garvan et al., 24 Oct 2025).

4. Combinatorial Interpretations

A hallmark of these third-order mock theta functions is their deep connection to partition theory. For example, $f(q)$ generates the parity difference of partitions: $$f(q) = \sum_{n=0}^{\infty} (p_{\mathrm{even}}(n) - p_{\mathrm{odd}}(n)) q^n$$ where $p_{\mathrm{even}}(n)$ (resp. $p_{\mathrm{odd}}(n)$) is the number of partitions of $n$ into an even (resp. odd) number of parts (Bringmann et al., 2011, Chen et al., 2010).

For $\omega(q)$ and $\nu(q)$, combinatorial interpretations involve partitions with bounded large parts: \begin{align*} \omega(q) &: \text{partitions with every odd part } < 2 \times (\text{smallest part}) \ \nu(-q) &: \text{distinct partitions with every odd part } < 2 \times (\text{smallest part}) \ \phi(q) &: \text{distinct partitions with each even part} \le 2 \times (\text{smallest part}) \end{align*} These results underlie generating-function relations and smallest parts congruences for the corresponding partition classes (Andrews et al., 2015, Baruah et al., 2019).

Additional partition statistics, such as rank differences and two-color partitions, have canonical generating functions involving $f(q)$ and $\phi(q)$, including novel identities equating numbers of partitions with positive odd rank and combinatorially defined two-color partitions (Andrews et al., 13 Jan 2025).

5. Analytic Continuation, Partial Theta, and Quantum Modularity

When analytically continued outside the unit disk ($|q|>1$), the third-order mock theta series express as combinations of partial theta functions (half-lattice sums) with $q$-hypergeometric error terms. For instance,

$f(q^{-1}) = 2\, v(q) - S(q)$

with $v(q)$ and $S(q)$ explicit partial theta series (Bringmann et al., 2011). This dichotomy—genuine mock behavior in $|q|<1$, partial theta in $|q|>1$—establishes the analytic bridge between mock modular forms and false (partial) theta functions.

Recent advances identify "antiquantum" $q$-series identities valid at roots of unity on $|q|=1$, relating mock theta functions to finite truncations of divergent $q$-hypergeometric companions. These identities refine Ramanujan's asymptotic radial-limit definitions and provide finite formulas for mock theta values at roots of unity, extending the viewpoint of quantum modularity (Folsom et al., 4 Jul 2025, Schneider, 2016).

6. Representation-Theoretic and Automorphic Embedding

Ramanujan's third-order mock theta functions naturally inhabit the representation-theoretic setting of mock Jacobi forms. Each is realized as (a specialization of) an optimal mock Jacobi theta function, with associated index and genus-zero symmetry group structure as classified by Cheng and Duncan (Cheng et al., 2016). The Fourier coefficients of these forms admit transcendental formulas in terms of traces of singular moduli.

Moreover, the vector-valued modularity of these functions under $SL_2(\mathbb{Z})$—explicitly computed by Watson and rederived using modern Appell–Lerch and Lerch sum techniques—confirms that appropriate completions transform as harmonic Maass-Jacobi forms, placing the mock theta functions at the intersection of analytic number theory, modular representation theory, and quantum topology (Garvan et al., 24 Oct 2025, Mortenson, 2012).

7. Contemporary Developments and Identities

Ongoing research uncovers new Hecke-type double-sum expansions, connections to Witten–Reshetikhin–Turaev invariants, and Beck-type companion partition identities generalizing Euler's pentagonal theorem (Mortenson, 2012, Ballantine et al., 2022). Sophisticated analytic techniques, including Poincaré series constructions, Bailey pairs, and modular traces, continue to unravel the structure of third-order mock theta coefficients and their congruence phenomena (Cui et al., 2018, Andersen et al., 2 Sep 2025, Garvan, 2014).

These developments reveal the unifying apparatus behind Ramanujan's third-order mock theta functions—a synergy of $q$-hypergeometric analysis, explicit modular transformation formulas, rich combinatorial meaning, and deep connections to the algebraic and automorphic theory of modular forms.

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References:

• Universal mock theta double sums and identities (Garvan, 2014)
• Rademacher-type expansions for Fourier coefficients (Andersen et al., 2 Sep 2025)
• Mock modular completion and automorphic properties (Bringmann et al., 2011, Garvan et al., 24 Oct 2025, Rhoades, 2011)
• Partition and combinatorial interpretations (Chen et al., 2010, Andrews et al., 13 Jan 2025, Ballantine et al., 2022, Andrews et al., 2015, Baruah et al., 2019)
• Quantum/antiquantum phenomena at roots of unity (Folsom et al., 4 Jul 2025, Schneider, 2016)
• Hecke/Jacobi embedding and Moonshine (Cheng et al., 2016)
• Hecke-type double sums and vector-valued modularity (Mortenson, 2012, Cui et al., 2018)