Cubic Analogues of the Jacobian Theta Function θ(z, q)

1993 ◽  
Vol 45 (4) ◽  
pp. 673-694 ◽  
Author(s):  
Michael Hirschhorn ◽  
Frank Garvan ◽  
Jon Borwein
Keyword(s):  
Hypergeometric Function ◽  
Theta Function ◽  
Modular Forms ◽  
Elliptic Function ◽  
Triple Product ◽  
Product Identity ◽  
Jacobi Triple Product Identity

AbstractThere are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity

scholarly journals Consequences of a sextuple-product identity

1987 ◽  
Vol 10 (3) ◽  
pp. 545-549
Author(s):  
John A. Ewell
Keyword(s):  
Triple Product ◽  
Product Identity ◽  
Special Cases ◽  
Jacobi Triple Product Identity

A sextuple-product identity, which essentially results from squaring the classical Gauss-Jacobi triple-product identity, is used to derive two trigonometrical identities. Several special cases of these identities are then presented and discussed.

scholarly journals A Family of Theta-Function Identities Based upon Combinatorial Partition Identities Related to Jacobi’s Triple-Product Identity

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 918
Author(s):  
Hari Mohan Srivastava ◽  
Rekha Srivastava ◽  
Mahendra Pal Chaudhary ◽  
Salah Uddin
Keyword(s):  
Theta Function ◽  
Subject Matter ◽  
Open Problem ◽  
Triple Product ◽  
Partition Identities ◽  
Product Identity ◽  
Recent Developments ◽  
The Subject ◽  
Theta Function Identities ◽  
R Functions

The authors establish a set of six new theta-function identities involving multivariable R-functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper, we consider and relate the multivariable R-functions to several interesting q-identities such as (for example) a number of q-product identities and Jacobi’s celebrated triple-product identity. Various recent developments on the subject-matter of this article as well as some of its potential application areas are also briefly indicated. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities and present a presumably open problem.

Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity

Integers ◽  
2011 ◽  
Vol 11 (6) ◽  
Author(s):  
Chuanan Wei ◽  
Dianxuan Gong
Keyword(s):  
Triple Product ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Jacobi Triple Product Identity ◽  
Liouville’S Theorem

AbstractBy means of Liouville's theorem, we show that Euler's pentagonal number theorem implies the Jacobi triple product identity.

scholarly journals Jacobi’s triple product, mock theta functions, unimodal sequences and the q-bracket

2018 ◽  
Vol 14 (07) ◽  
pp. 1961-1981
Author(s):  
Robert Schneider
Keyword(s):  
Unit Circle ◽  
Theta Function ◽  
Modular Forms ◽  
Statistical Physics ◽  
Theta Functions ◽  
Triple Product ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Quantum Modular Forms ◽  
Unimodal Sequences

In Ramanujan’s final letter to Hardy, he listed examples of a strange new class of infinite series he called “mock theta functions”. It turns out all of these examples are essentially specializations of a so-called universal mock theta function [Formula: see text] of Gordon–McIntosh. Here we show that [Formula: see text] arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms—under the action of an operator related to statistical physics and partition theory, the [Formula: see text]-bracket of Bloch–Okounkov. Second, we find [Formula: see text] to extend in [Formula: see text] to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other [Formula: see text]-hypergeometric series. Finally, we look at interesting “quantum” behaviors of mock theta functions inside, outside, and on the unit circle.

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