Bailey lemma
E440253
The Bailey lemma is a key result in the theory of basic hypergeometric series that provides a systematic method for generating Rogers–Ramanujan-type identities and other q-series relations.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Bailey lemma canonical | 3 |
| well-poised Bailey lemma | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4437401 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bailey lemma Context triple: [Rogers–Ramanujan-type identities, relatedTo, Bailey lemma]
-
A.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
B.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
C.
Fitting lemma
The Fitting lemma is a result in group theory and module theory that characterizes how certain algebraic structures decompose into direct sums of invariant subcomponents, often involving nilpotent and invertible parts.
-
D.
Fatou's lemma
Fatou's lemma is a fundamental result in measure theory that provides an inequality relating the integral of the pointwise limit inferior of a sequence of nonnegative measurable functions to the limit inferior of their integrals.
-
E.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bailey lemma Target entity description: The Bailey lemma is a key result in the theory of basic hypergeometric series that provides a systematic method for generating Rogers–Ramanujan-type identities and other q-series relations.
-
A.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
B.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
C.
Fitting lemma
The Fitting lemma is a result in group theory and module theory that characterizes how certain algebraic structures decompose into direct sums of invariant subcomponents, often involving nilpotent and invertible parts.
-
D.
Fatou's lemma
Fatou's lemma is a fundamental result in measure theory that provides an inequality relating the integral of the pointwise limit inferior of a sequence of nonnegative measurable functions to the limit inferior of their integrals.
-
E.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical lemma
ⓘ
result in q-series theory ⓘ tool in basic hypergeometric series ⓘ |
| appliesTo |
convergent q-series
ⓘ
formal power series in q ⓘ |
| category | q-series identity machinery ⓘ |
| field | mathematics ⓘ |
| generalizes | classical hypergeometric transformations to q-setting ⓘ |
| hasConcept |
Bailey chain
ⓘ
Bailey pair ⓘ
linked to:
Bailey pairs
Bailey transform ⓘ |
| hasFormulation | relation between two sequences forming a Bailey pair ⓘ |
| hasVariant |
bilateral Bailey lemma
ⓘ
limiting form of Bailey lemma ⓘ well-poised Bailey lemma ⓘ
linked to:
Bailey lemma
|
| implies |
Andrews–Gordon identities
ⓘ
linked to:
Rogers–Ramanujan-type identities
Rogers–Ramanujan identities ⓘ many Rogers–Ramanujan-type partition theorems ⓘ |
| influenced |
George E. Andrews
ⓘ
subsequent work on partition identities ⓘ |
| introducedBy | W. N. Bailey ⓘ |
| involves |
basic hypergeometric series notation
ⓘ
q-Pochhammer symbol ⓘ q-shifted factorials ⓘ |
| namedAfter | W. N. Bailey ⓘ |
| relatedTo |
Andrews–Gordon identities
ⓘ
linked to:
Rogers–Ramanujan-type identities
Bailey lattice ⓘ Rogers–Fine identity ⓘ Rogers–Ramanujan identities ⓘ q-binomial theorem ⓘ |
| relates | Bailey pairs relative to a parameter a ⓘ |
| subfield |
analytic number theory
ⓘ
basic hypergeometric series ⓘ combinatorics ⓘ q-series ⓘ |
| typicalConclusion | produces a new Bailey pair (α'_n, β'_n) relative to the same parameter ⓘ |
| typicalHypothesis | given a Bailey pair (α_n, β_n) relative to a ⓘ |
| usedFor |
constructing infinite families of q-identities
ⓘ
deriving q-series identities ⓘ generating Rogers–Ramanujan-type identities ⓘ producing Bailey chains ⓘ producing Bailey pairs ⓘ proving partition identities ⓘ transforming basic hypergeometric series ⓘ |
| usedIn |
combinatorial proofs of partition theorems
ⓘ
proofs of modular-type q-series identities ⓘ theory of mock theta functions ⓘ |
| yearIntroduced | 1940s ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Bailey lemma Description of subject: The Bailey lemma is a key result in the theory of basic hypergeometric series that provides a systematic method for generating Rogers–Ramanujan-type identities and other q-series relations.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bailey lemma