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.

scholarly journals A family of theta-function identities based upon Rα,Rβ and Rm-functions related to Jacobi’s triple-product identity

2020 ◽  
Vol 108 (122) ◽  
pp. 23-32
Author(s):  
Mahendra Chaudhary
Keyword(s):  
Theta Function ◽  
Open Problem ◽  
Continued Fraction ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Theta Function Identities ◽  
Jacobi's Triple Product Identity

We establish a set of two new relationships involving R?,R? and Rm-functions, which are based on Jacobi?s famous triple-product identity. We, also provide answer for an open problem of Srivastava, Srivastava, Chaudhary and Uddin, which suggest to find an inter-relationships between R?,R? and Rm(m ? N), q-product identities and continued-fraction identities.

scholarly journals Some Theta-Function Identities Related to Jacobi’s Triple-Product Identity

2018 ◽  
Vol 11 (1) ◽  
pp. 1 ◽  
Author(s):  
Hari M. Srivastava ◽  
M. P. Chaudhary ◽  
Sangeeta Chaudhary
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Theta Function Identities ◽  
Jacobi's Triple Product Identity

The main object of this paper is to present some q-identities involving some of the theta functions of Jacobi and Ramanujan. These q-identities reveal certain relationships among three of the theta-type functions which arise from the celebrated Jacobi’s triple-product identity in a remarkably simple way. The results presented in this paper are motivated by some recent works by Chaudhary et al. (see [4] and [5]) and others (see, for example, [1] and [13]).

scholarly journals Relations between Rα, Rβ and Rm functions related to Jacobi’s triple-product identity and the family of theta-function identities

2021 ◽  
Vol 27 (2) ◽  
pp. 1-11
Author(s):  
M. P. Chaudhary ◽  
Keyword(s):  
Theta Function ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
The Family ◽  
Open Question ◽  
Theta Function Identities ◽  
Jacobi's Triple Product Identity

In this paper, the author establishes a set of three new theta-function identities involving Rα, Rβ and Rm 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 answer a open question of Srivastava et al [33], and established relations in terms of Rα, Rβ and Rm (for m = 1, 2, 3), and q-products identities. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities.

Partition identities arising from Somos’s theta function identities

2016 ◽  
Vol 63 (2) ◽  
pp. 303-313 ◽  
Author(s):  
B. R. Srivatsa Kumar ◽  
R. G. Veeresha
Keyword(s):  
Theta Function ◽  
Partition Identities ◽  
Theta Function Identities

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 Unification of the Quintuple and Septuple Product Identities

10.37236/1008 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wenchang Chu ◽  
Qinglun Yan
Keyword(s):  
Functional Equation ◽  
Theta Function ◽  
General Equation ◽  
Triple Product ◽  
Equation Method ◽  
Jacobi Theta Function ◽  
Common Generalization ◽  
Product Identity ◽  
The Common ◽  
Free Parameters

By combining the functional equation method with Jacobi's triple product identity, we establish a general equation with five free parameters on the modified Jacobi theta function, which can be considered as the common generalization of the quintuple, sextuple and septuple product identities. Several known theta function formulae and new identities are consequently proved.

scholarly journals Partition identities arising from theta function identities

2008 ◽  
Vol 24 (6) ◽  
pp. 955-970 ◽  
Author(s):  
Nayandeep Deka Baruah ◽  
Bruce C. Berndt
Keyword(s):  
Theta Function ◽  
Partition Identities ◽  
Theta Function Identities

Words

PMLA ◽  
1935 ◽  
Vol 50 (4) ◽  
pp. 1320-1327
Author(s):  
Colbert Searles
Keyword(s):  
Subject Matter ◽  
Assistant Professor ◽  
The Subject

THE germ of that which follows came into being many years ago in the days of my youth as a university instructor and assistant professor. It was generated by the then quite outspoken attitude of colleagues in the “exact sciences”; the sciences of which the subject-matter can be exactly weighed and measured and the force of its movements mathematically demonstrated. They assured us that the study of languages and literature had little or nothing scientific about it because: “It had no domain of concrete fact in which to work.” Ergo, the scientific spirit was theirs by a stroke of “efficacious grace” as it were. Ours was at best only a kind of “sufficient grace,” pleasant and even necessary to have, but which could, by no means ensure a reception among the elected.

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