Bailey Chain in q-Series and Modular Forms

• Bailey chain is a systematic method that iteratively transforms Bailey pairs to produce diverse q-series identities, including Rogers–Ramanujan type identities.
• It employs transformation formulas and change-of-base techniques to isolate pure mock modular components from mixed analytic structures.
• Its applications span combinatorics, analytic number theory, and physics, linking partition statistics to mock theta functions and modular identities.

The Bailey chain is a systematic method for generating infinite families of $q$-series identities from Bailey pairs, with applications spanning modular forms, partition theory, and mock theta functions. Originating from Andrews' extension of Bailey's lemma, the Bailey chain operates by iteratively transforming Bailey pairs—sequences connected by $q$-hypergeometric summations—yielding new multisum expressions and facilitating the discovery of identities including those of Rogers–Ramanujan type and deep constructions in the theory of mock modular forms.

1. Structure and Iteration of Bailey Chains

A Bailey pair relative to a parameter $a$ consists of sequences $\{\alpha_n\}$ and $\{\beta_n\}$ such that

$$\beta_n = \sum_{k=0}^n \frac{\alpha_k}{(q)_{n-k}\,(a q)_{n+k}},$$

with $(a)_n = (a;q)_n = \prod_{j=0}^{n-1} (1 - a q^j)$ standard $q$-hypergeometric notation. Given such a pair, new Bailey pairs can be constructed using transformation formulas: $$\alpha'_n = \frac{(b)_n (c)_n (a q/bc)^n}{(a q/b)_n (a q/c)_n} \, \alpha_n,$$

$$\beta'_n = \sum_{k=0}^n \frac{(b)_k (c)_k (a q/bc)_{n-k} (a q/bc)^k}{(q)_{n-k}(a q/b)_n (a q/c)_n}\,\beta_k.$$

Iterating these transformations defines the Bailey chain, producing a sequence of Bailey pairs and corresponding $q$-series identities of increasing complexity. Specializing parameters in these iterations yields classical results such as the Rogers–Ramanujan identities.

2. Bailey Chain and Mock Theta Functions

While classical Bailey chains naturally generate mixed mock modular objects (i.e., series with both modular and mock modular summands), their application to particular Bailey pairs allows the construction of multisums that exhibit pure mock modular behavior. For instance, applying the Bailey chain to

$$\alpha_n = \begin{cases} 1, & n=0, \ (-1)^n q^{n(n+1)/2}(1+q^n), & n>0, \end{cases}$$

and $\beta_n = \frac{(-q)_n}{(q)_n}$, successive chain iterations produce multisums expressible in terms of Appell–Lerch sums. This approach is fundamental in connecting iterative $q$-hypergeometric multisums to known mock theta functions, as in Watson’s 3rd-order function via such transformations. The Bailey chain thus serves as a bridge between combinatorial $q$-series and the analytic structures underlying mock modular forms (Lovejoy et al., 2012).

3. Change of Base: Isolation of Mock Modularity

A technique critical for isolating mock modular components within Bailey chains is the change of base, specifically as formulated by Bressoud–Ismail–Stanton: $$\alpha'_n = \frac{1+q}{1+q^{2n+1}} q^{-n}\alpha_n(q^2), \quad \beta'_n = \sum_{k=0}^n \frac{(-q^2)_{n-k}}{(q^2;q^2)_{n-k}}q^k\beta_k.$$ This change repackages Bailey pairs from base $q$ to base $q^2$, crucial for constructing multisums whose analytic continuation yields pure mock theta functions, as opposed to mixed modular forms. The authors of (Lovejoy et al., 2012) utilize this transformation to generate explicit families of $q$-hypergeometric multisums, which align exactly (up to weakly holomorphic modular forms) with classical mock theta functions via Appell–Lerch sum representations. This method is essential not only for constructing mock theta functions but also for exposing the underlying indefinite quadratic forms inherent in their modularity.

4. Identification of Multisum and Classical Mock Theta Identities

The Bailey chain, enhanced by change-of-base transformations, generates explicit multisums

$$R_k(q) = \sum_{n_k \ge n_{k-1} \ge \cdots \ge n_1 \ge 0} q^{\text{quadratic form in } n_i} B_k(n_k, \dots, n_1; q),$$

where $B_k(\cdot)$ involves products of $q$-Pochhammer symbols and combinatorial factors (formulated explicitly in [(Lovejoy et al., 2012), eqs. (1.14)–(1.17)]). Identities such as

$$R_3(q) = v(-q), \qquad R_4(q) = -\phi(q^4) + M_1(q),$$

link multisum constructions directly to classical mock theta functions $v(q)$ and $\phi(q)$, and $M_1(q)$ denoting a weakly holomorphic modular form. These results provide not only new expressions for mock theta functions but also further combinatorial interpretations and avenues for asymptotic analysis or congruence investigation.

5. Implications for Modular and Combinatorial Structures

The expansion of the Bailey chain into the field of mock modularity has significant ramifications:

• It extends the classical role of Bailey chains—from producing Rogers–Ramanujan-type identities to generating families of mock theta functions and mixed mock modular forms.
• The established connection between $q$-hypergeometric multisums and Appell–Lerch sums advances the understanding of mock theta functions as holomorphic parts of harmonic weak Maass forms, reinforcing Zwegers’ framework.
• Proven identities suggest further combinatorial, modular, and even physical implications (e.g., connections to black hole entropy and wall-crossing phenomena), and enable new combinatorial interpretations for partition statistics encoded in these multisums.
• The methodology highlights the potential to discover additional families of mock modular forms and combinatorial identities by judicious application of change-of-base techniques to other Bailey pairs.

6. Broader Applications and Future Directions

The Bailey chain, especially when merged with changes of base and explicit identification of multisums as mock theta functions, is a unifying mechanism across multiple fields:

• Combinatorics: partition statistics (ranks, cranks, spt-functions) derive new expressions and congruences from Bailey chain constructions.
• Analytic number theory: the chain reveals structural hierarchies among $q$-series, modular forms, and recently mock modular forms.
• Representation theory and physics: Bailey chain multisum identities relate to characters of affine Lie algebras, vertex operator algebras, and spectral sums in statistical mechanics.
• Further research may focus on extending the chain’s scope to higher-dimensional analogues, more general quadratic forms, and broader families of modular objects, leveraging results from change-of-base techniques and the interplay with indefinite theta series.

Summary Table: Bailey Chain Effects

Bailey Chain Application	Resulting Object	Technique Required
Standard iteration	Mixed mock modular forms	Iterative Bailey lemma
Change of base (Bressoud et al.)	Genuine mock theta functions	Switch base $q \to q^2$
Multisum to Appell–Lerch sum	Explicit representation	Series transformation
Multisum–modular identity	Mock theta/congruences	Modular completion

In conclusion, the Bailey chain is a recursive device that systematically generates multisum $q$-series identities, with change-of-base playing a crucial role in isolating pure mock modularity. The approach establishes explicit links between $q$-hypergeometric multisums and classical mock theta functions, empowers new combinatorial and modular identities, and opens wide prospects for both analytic and combinatorial advances.