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.

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 Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Analogous Result ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Nonnegative Coefficients ◽  
Jacobi's Triple Product Identity

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

scholarly journals A simple proof of an identity of Ramanujan

1983 ◽  
Vol 34 (1) ◽  
pp. 31-35 ◽  
Author(s):  
M. D. Hirschhorn
Keyword(s):  
Simple Proof ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Jacobi's Triple Product Identity

AbstractOne of Ramanujan's unpublished, unproven identities has excited considerable interest over the years. Indeed, no fewer than four proofs have appeared in the literature. The object of this note is to present yet another proof, simpler than the others, relying only on Jacobi's triple product identity.

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.

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