Bailey Pairs in q-Series and Modular Forms

• Bailey pairs are defined as sequence pairs (αₙ, βₙ) linked by a q-Pochhammer relation, crucial for formulating q-series and partition identities.
• The Bailey transform and conjugate pairs enable the derivation of two-variable Hecke–Rogers identities, directly impacting mock theta and universal mock modular functions.
• Iterative applications via the Bailey chain generate infinite families of combinatorial identities, bridging analytic methods with modular form theory.

A Bailey pair is a pair of sequences $(\alpha_n, \beta_n)$ (relative to a base parameter $a$) that satisfy the relation

$$\beta_n = \sum_{r=0}^{n} \frac{\alpha_r}{(q;q)_{n-r} (a q; q)_{n+r}}$$

where $(x;q)_n$ denotes the $q$-Pochhammer symbol. Bailey pairs, and the Bailey transform that connects them to conjugate Bailey pairs, are fundamental tools in the analytic theory of $q$-series identities, especially those concerned with mock theta functions and Rogers–Ramanujan–type relations. Much of the modern structure surrounding partition identities, mock modular forms, and basic hypergeometric series transformations is built upon the manipulation and iteration of Bailey pairs, most notably via Bailey’s lemma, the Bailey transform, and its generalizations.

1. Bailey Transform and Its Role

The Bailey transform is a bilinear identity linking a Bailey pair $(\alpha_n, \beta_n)$ and a conjugate Bailey pair $(\gamma_n, \delta_n)$. In the normalized form given in [(Ji et al., 2014), Theorem 2.1], if $(\alpha_n, \beta_n)$ is a Bailey pair relative to $a$ (i.e., $$\beta_n = \sum_{r=0}^n \frac{\alpha_r}{(q;q)_{n-r} (a q;q)_{n+r}}$$) and $(\gamma_n, \delta_n)$ is a conjugate Bailey pair relative to $a$ (i.e., $$\gamma_n = \sum_{r=n}^{\infty} \frac{\delta_r}{(q;q)_{r-n}(a q;q)_{r+n}}$$), then: $$\sum_{n=0}^\infty \alpha_n \gamma_n = \sum_{n=0}^\infty \beta_n \delta_n$$ This transform allows direct passage between $q$-series involving different (but related) summands and often provides a mechanism to derive nontrivial identities for mock theta functions, partition ranks, and related objects.

In (Ji et al., 2014), a critical ingredient is a conjugate Bailey pair due to Warnaar (Lemma 2.2 in the paper), which is expressed (with $a$ and $b$ as parameters, later specialized to $a = z$, $b = z^{-1}q$ for mock theta applications) as

$\begin{aligned} o_n &= \frac{(a q, b, q;q)_n}{(ab; q)_{2n}}\,q^{-n/2} \ I_n &= \frac{1-abq^{2n}}{1-ab} \cdot \{\text{series in %%%%18%%%%, %%%%19%%%%, %%%%20%%%%}\} \end{aligned}$

This pair, when conjugated suitably, captures the two-variable aspect necessary for identities involving universal mock theta functions such as $g_2(z;q)$, $g_3(z;q)$, and $K(z;q)$.

Multiple Bailey pairs are employed, some arising from special cases of Slater's $_6\psi_6$ identity, with $\alpha_n$ and $\beta_n$ sequences containing highly nontrivial expressions, including alternating signs, powers of $q$, and reciprocal factors such as $(1 - q^{6n+1})^{-1}$. Each such pair is inserted into the Bailey transform to "match" the structure of particular mock or partition function identities.

2. Two-Variable Hecke–Rogers Identities Derived from Bailey Pairs

The central technical achievement is a direct proof of Garvan’s two-variable Hecke–Rogers identities using the Bailey transform. These identities relate double sums indexed by $n$ and an inner index (often $j$) to natural generating functions for mock theta objects.

For example, for the Dyson rank function $R(z;q)$ (equivalent, up to normalization, to $g_3(z;q)$), one has

$$(zq, z^{-1}q, q; q)_\infty R(z;q) = \sum_{n=0}^\infty \sum_{j=0}^{\lfloor n/2 \rfloor} (-1)^{n+j} \big( z^{n - 3j} + z^{3j-n+1} \big) q^{n^2 - 3j^2 + n - j}$$

The mechanics involve expressing the generating function for $R(z;q)$ via input of the relevant Bailey pair into the Bailey transform, matched with the Warnaar conjugate pair.

Similarly, for the universal mock theta function $g_2(z;q)$ (see Theorem 1.2 in the paper), the identity is

$(1-z)(zq, z^{-1}q, q; q)_\infty g_2(z;q) = \cdots$

where the right side unfolds into a combination of two double sums, each with explicit $q$-exponents and controlled sign patterns.

Critically, these constructions are not isolated: the Bailey chain (Theorem 2.6) provides a systematic way to generate infinite families of such identities, indexed by a parameter $k$, by iteratively applying the Bailey transform and inserting conjugate pairs with carefully chosen parameters. This yields compact double-sum formulations and, through index rearrangements, recovers numerous forms of the classical identities.

3. Structure and Construction of Bailey Pairs Used

Several types of Bailey pairs are constructed, some with features tailored for inserting into the Bailey transform with the Warnaar conjugate pair:

Lemma / Pair Form	Parameter	$\alpha_n$ structure	$\beta_n$ structure
Lemma 2.3	relative to $q$	$(-1)^n q^{n^2} / (1-q^{6n+1})$	$(q;q)_{2n}$ in denominator
Lemma 2.4	relative to $q^2$	vanishing for odd $n$	$(-q;q)_n$, $Q_{2n+1}=0$
Lemma 2.5	relative to $q$	$(-1)^n q^{(*)}$, denom $(1-q^{2n+1})$

Each pair is characterized by alternating sign structure, quadratic or mixed degree $q$-powers, and vanishing properties controlled by parameter choices (e.g., for even/odd $n$).

The Warnaar conjugate is parameterized so that, upon specialization, the double-variable nature of the mock theta function is built directly into its sequence.

4. Extension to Infinite Families and the Bailey Chain

The application of the Bailey chain mechanism, as in Theorem 2.6, enables the extension of the Hecke–Rogers identities to infinite families. The mechanism can be abstractly described as:

• Start with a Bailey pair relative to parameter $a$;
• At each step, apply transformation rules (given explicitly in the paper) to generate a new Bailey pair with higher complexity or higher "tier" in the chain;
• After each iteration, insert the new pair into the Bailey transform, possibly accompanied by a suitably specialized conjugate pair;
• The outcome is a "family" of double-sum identities with additional combinatorial or analytic structure, closed under the chain operation.

The compactness and generality of the resulting identities (e.g., equations (1.15)–(1.16)) display deeper underlying symmetries and invite modular or automorphic interpretations. This iterativity is a distinguishing feature of the method relative to classical, more ad hoc, proofs.

5. Implications for the Theory of Mock Theta Functions

The proofs obtained via Bailey pairs and the Bailey transform provide a direct analytic route to Garvan's two-variable Hecke–Rogers identities, replacing previous approaches based on hypergeometric summations. The methodology demonstrates that:

• Bailey pair technology enables systematic derivation of identities connecting the combinatorics of partitions (notably ranks and overpartition ranks) with analytic $q$-series representations of mock theta functions;
• The fact that the universal mock theta functions $g_2(z;q)$ and $g_3(z;q)$ serve as "master objects" for many of Ramanujan's mock theta functions underscores the combinatorial depth revealed by Bailey-theoretic proofs;
• The emergence of infinite families suggests the existence of rich hierarchies in the analytic and modular structure of these objects, opening directions for discovering further transformations, symmetries, or hierarchical modularity;
• Prospective applications include modular and automorphic lifts, detailed combinatorial interpretations of subsums, and explorations of connections with quantum modular forms and non-commutative geometry.

Potential extensions noted in the paper include the search for interpretations of the infinite double/multisums, generalization to additional families, and deepening the modular or quantum modular implications of the two-variable Hecke–Rogers identities.

6. Technical Summation: Key Formulas and the Use of Product Identities

The proofs and constructions rely crucially on $q$-Pochhammer product identities, most notably Jacobi’s triple product. In particular: $$(q, zq, q/z;q)_\infty = \sum_{n=-\infty}^\infty (-1)^n z^n q^{n(n+1)/2}$$ This identity is repeatedly invoked after transforming double sums or reorganizing terms, to rewrite series in compact product form or to compare with known modular objects. Generalizations of Jacobi's identity also appear in the verification of the conjugate pair properties.

Every derived Hecke–Rogers identity is ultimately expressible in a form where both sum and product sides can be directly compared, enabling modularity checks and analytic continuation arguments.

7. Summary and Outlook

By leveraging a suite of Bailey pairs—including the Warnaar conjugate and pairs drawn from Slater's $\,_6\psi_6$ specializations—and plugging them systematically into the Bailey transform, the work provides compact, directly combinatorial proofs of sophisticated two-variable Hecke–Rogers identities for the universal mock theta functions. The resulting identities, notably in the compact forms for $g_3(z;q)$ and $g_2(z;q)$, not only reaffirm known results (e.g., those by Garvan) but also display an extensibility to infinite families mediated by the Bailey chain.

The approach opens pathways for further development in the analytic theory of partitions, the structure of mock modular forms, and the discovery of new modular and quantum modular phenomena, all rooted in the algebraic combinatorics of Bailey pair technology (Ji et al., 2014).