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 (4)

Label Occurrences
Jacobi triple product canonical 3
Jacobi theta functions 1
Jacobi's triple product identity 1

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 ⓘ

Referenced by (6)

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

Carl Gustav Jacob Jacobi → notableWork → Jacobi triple product ⓘ
this entity surface form: Jacobi theta functions
Carl Gustav Jacob Jacobi → notableWork → Jacobi triple product ⓘ
Carl Gustav Jacob Jacobi → notableWork → Jacobi triple product ⓘ
this entity surface form: Jacobi’s triple product identity
Jacobi’s four-square theorem → relatedTo → Jacobi triple product ⓘ
Carl → notableWork → Jacobi triple product ⓘ
subject surface form: Carl Gustav Jacob Jacobi
Carl → notableWork → Jacobi triple product ⓘ
subject surface form: Carl Gustav Jacob Jacobi
this entity surface form: Jacobi's triple product identity