Jacobi triple product
E182749
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Jacobi triple product canonical | 3 |
| Jacobi theta functions | 1 |
| Jacobi's triple product identity | 1 |
| Jacobi’s triple product identity | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1615214 — 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: Jacobi triple product Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi triple product]
-
A.
Rogers–Ramanujan-type identities
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.
-
B.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
C.
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.
-
D.
arithmetic–geometric mean identities
Arithmetic–geometric mean identities are a collection of formulas and relationships that express various mathematical constants and special functions in terms of the arithmetic–geometric mean of two numbers.
-
E.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
- 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: Jacobi triple product Target entity description: The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
-
A.
Rogers–Ramanujan-type identities
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.
-
B.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
C.
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.
-
D.
arithmetic–geometric mean identities
Arithmetic–geometric mean identities are a collection of formulas and relationships that express various mathematical constants and special functions in terms of the arithmetic–geometric mean of two numbers.
-
E.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical identity
ⓘ
q-series identity ⓘ theta function identity ⓘ |
| appearsIn |
Jacobi’s theory of elliptic functions
ⓘ
q-analogs in special functions ⓘ |
| category |
Fourier series identity
ⓘ
infinite product identity ⓘ |
| conditionOnq | |q| < 1 ⓘ |
| domainOfq | complex numbers ⓘ |
| domainOfz | complex numbers ⓘ |
| field |
complex analysis
ⓘ
number theory ⓘ |
| generalizes | Euler’s identity for sine product ⓘ |
| givesProductRepresentationOf |
Jacobi theta functions
ⓘ
surface form:
Jacobi theta function
|
| givesSeriesRepresentationOf |
Jacobi theta functions
ⓘ
surface form:
Jacobi theta function
|
| hasApplicationIn |
analytic number theory
ⓘ
combinatorial number theory ⓘ mathematical physics ⓘ |
| hasForm | sum-product identity ⓘ |
| hasInfiniteProductSide | ∏_{n=1}^{∞} (1 - q^{2n})(1 + z q^{2n-1})(1 + z^{-1} q^{2n-1}) ⓘ |
| hasInfiniteSumSide | ∑_{n=-∞}^{∞} z^{n} q^{n^{2}} ⓘ |
| hasqSeriesType | basic hypergeometric series ⓘ |
| hasVariable |
q
ⓘ
z ⓘ |
| implies |
Euler pentagonal number theorem
ⓘ
Rogers–Ramanujan-type identities ⓘ
surface form:
Rogers–Ramanujan type identities
|
| isClassicalResult | 19th-century mathematics ⓘ |
| isToolFor |
manipulating q-series
ⓘ
studying modular transformations of theta functions ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| relates |
infinite product
ⓘ
infinite sum ⓘ q-series ⓘ theta functions ⓘ |
| usedIn |
combinatorics
ⓘ
quantum field theory ⓘ
surface form:
conformal field theory
elliptic functions ⓘ representation theory ⓘ string theory ⓘ theory of modular forms ⓘ theory of partitions ⓘ |
| usedToDerive |
generating functions for integer partitions
ⓘ
identities in basic hypergeometric series ⓘ product expansions of theta functions ⓘ |
| validFor | z ≠ 0 ⓘ |
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: Jacobi triple product Description of subject: The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Jacobi theta functions
this entity surface form:
Jacobi’s triple product identity
subject surface form:
Carl Gustav Jacob Jacobi
subject surface form:
Carl Gustav Jacob Jacobi
this entity surface form:
Jacobi's triple product identity