Jacobi triple product

E182749

mathematical identity q-series identity theta function identity

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.

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All labels observed (5)

Label	Occurrences
Jacobi triple product canonical	4
Jacobi triple product identity	2
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 function ⓘ linked to: Jacobi theta functions
givesSeriesRepresentationOf	Jacobi theta function ⓘ linked to: Jacobi theta functions
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 ⓘ linked to: 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 ⓘ conformal field theory ⓘ linked to: quantum 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 ⓘ