Rogers–Ramanujan-type identities
E95684
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
All labels observed (7)
| Label | Occurrences |
|---|---|
| Rogers–Ramanujan identities | 2 |
| Andrews–Gordon identities | 1 |
| Gordon identities | 1 |
| Göllnitz–Gordon identities | 1 |
| Rogers–Ramanujan type identities | 1 |
| Rogers–Ramanujan-type identities canonical | 1 |
| Schur identities | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T795893 — 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: Rogers–Ramanujan-type identities Context triple: [Dyson’s transform, relatedTo, Rogers–Ramanujan-type identities]
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A.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
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B.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
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C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
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D.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
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E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- 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: Rogers–Ramanujan-type identities Target entity description: Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
A.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
-
B.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
C.
Conway’s topograph
Conway’s topograph is a geometric visualization tool introduced by mathematician John H. Conway to study binary quadratic forms and their arithmetic properties using a planar graph of curves and regions.
-
D.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
partition identity ⓘ q-series identity ⓘ |
| field |
combinatorics
ⓘ
number theory ⓘ q-series ⓘ theory of modular forms ⓘ |
| generalizes |
Rogers–Ramanujan-type identities
self-linksurface differs
ⓘ
surface form:
Rogers–Ramanujan identities
|
| hasGeneralizationMethod |
Andrews–Baxter–Forrester method
ⓘ
Bailey chain method ⓘ vertex operator method ⓘ |
| hasProperty |
combinatorial
ⓘ
deep ⓘ modular ⓘ q-analytic ⓘ |
| namedAfter |
Leonard James Rogers
ⓘ
Srinivasa Ramanujan ⓘ |
| relatedTo |
Rogers–Ramanujan-type identities
self-linksurface differs
ⓘ
surface form:
Andrews–Gordon identities
Bailey chains ⓘ Bailey lemma ⓘ Bailey pairs ⓘ Rogers–Ramanujan-type identities self-linksurface differs ⓘ
surface form:
Gordon identities
Rogers–Ramanujan-type identities self-linksurface differs ⓘ
surface form:
Göllnitz–Gordon identities
Rogers–Ramanujan-type identities self-linksurface differs ⓘ
surface form:
Schur identities
Slater’s list of identities ⓘ affine Lie algebras ⓘ basic hypergeometric series ⓘ conformal field theory ⓘ generating functions ⓘ infinite product expansions ⓘ integer partitions with difference conditions ⓘ mock theta functions ⓘ modular equations ⓘ modular functions ⓘ modular invariance ⓘ partition theory ⓘ q-hypergeometric series ⓘ q-product expansions ⓘ representation theory ⓘ restricted partition functions ⓘ theta functions ⓘ vertex operator algebras ⓘ |
| usedIn |
combinatorial bijections
ⓘ
proofs of partition congruences ⓘ q-series transformations ⓘ study of modular forms of half-integral weight ⓘ |
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: Rogers–Ramanujan-type identities Description of subject: Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Dyson’s transform
this entity surface form:
Rogers–Ramanujan identities
Rogers–Ramanujan-type identities
→
generalizes
→
Rogers–Ramanujan-type identities
self-linksurface differs
ⓘ
this entity surface form:
Rogers–Ramanujan identities
Rogers–Ramanujan-type identities
→
relatedTo
→
Rogers–Ramanujan-type identities
self-linksurface differs
ⓘ
this entity surface form:
Andrews–Gordon identities
Rogers–Ramanujan-type identities
→
relatedTo
→
Rogers–Ramanujan-type identities
self-linksurface differs
ⓘ
this entity surface form:
Göllnitz–Gordon identities
Rogers–Ramanujan-type identities
→
relatedTo
→
Rogers–Ramanujan-type identities
self-linksurface differs
ⓘ
this entity surface form:
Gordon identities
Rogers–Ramanujan-type identities
→
relatedTo
→
Rogers–Ramanujan-type identities
self-linksurface differs
ⓘ
this entity surface form:
Schur identities
this entity surface form:
Rogers–Ramanujan type identities