Ramanujan theta function

E355433

mathematical function q-series special function

The Ramanujan theta function is a special type of q-series introduced by Srinivasa Ramanujan that plays a central role in the theory of modular forms, partitions, and mock theta functions.

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All labels observed (3)

Label	Occurrences
Ramanujan theta function canonical	1
Ramanujan’s fifth order mock theta function χ1(q)	1
Ramanujan’s seventh order mock theta functions	1

Statements (45)

Predicate	Object
instanceOf	mathematical function ⓘ q-series ⓘ special function ⓘ
appearsIn	Ramanujan’s lost notebook ⓘ Ramanujan’s lost notebook ⓘ surface form: Ramanujan’s notebooks
convergesFor	\|q\| < 1 ⓘ
dependsOn	complex parameter q ⓘ complex variable z ⓘ
domain	complex numbers ⓘ
field	complex analysis ⓘ modular forms ⓘ number theory ⓘ partition theory ⓘ q-series theory ⓘ
generalizationOf	certain Jacobi theta functions ⓘ
hasProperty	admits product expansions ⓘ given by an infinite series in powers of q ⓘ holomorphic in z for fixed q with \|q\| < 1 ⓘ satisfies functional equations ⓘ
hasVariable	q ⓘ z ⓘ
introducedBy	Srinivasa Ramanujan ⓘ
introducedIn	early 20th century ⓘ
namedAfter	Srinivasa Ramanujan ⓘ
relatedTo	Dedekind eta function ⓘ Jacobi theta functions ⓘ surface form: Jacobi theta function mock theta function ⓘ modular equations ⓘ modular forms ⓘ SL(2,ℤ) ⓘ surface form: modular group SL(2,Z) q-Pochhammer symbol ⓘ q-hypergeometric series ⓘ
studiedBy	Bruce C. Berndt ⓘ G. N. Watson ⓘ George E. Andrews ⓘ
usedIn	Ramanujan partition congruences ⓘ surface form: Ramanujan’s congruences for partition function modular transformation formulas ⓘ q-series identities ⓘ theory of mock theta functions ⓘ theory of modular forms ⓘ theory of partitions ⓘ
usedToDefine	many mock theta functions ⓘ
usedToDerive	q-series transformation formulas ⓘ theta-type identities ⓘ
usedToExpress	partition generating functions ⓘ

Referenced by (3)

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

Srinivasa Ramanujan → notableWork → Ramanujan theta function ⓘ

mock theta functions → hasExample → Ramanujan theta function ⓘ

this entity surface form: Ramanujan’s fifth order mock theta function χ1(q)

mock theta functions → hasExample → Ramanujan theta function ⓘ

this entity surface form: Ramanujan’s seventh order mock theta functions