Rogers–Ramanujan-type identities

E95684

mathematical concept partition identity q-series identity

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.

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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]

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
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 identities ⓘ linked to: Rogers–Ramanujan-type 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	Andrews–Gordon identities ⓘ linked to: Rogers–Ramanujan-type identities Bailey chains ⓘ Bailey lemma ⓘ Bailey pairs ⓘ Gordon identities ⓘ linked to: Rogers–Ramanujan-type identities Göllnitz–Gordon identities ⓘ linked to: Rogers–Ramanujan-type identities Schur identities ⓘ linked to: Rogers–Ramanujan-type 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

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Dyson’s transform → relatedTo → Rogers–Ramanujan-type identities ⓘ

subject linked to: Dyson’s transform in number theory

Srinivasa Ramanujan → notableWork → Rogers–Ramanujan identities ⓘ