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 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 Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Matthieu Josuat-Vergès ◽  
Jang-Soo Kim
Keyword(s):  
Functional Equation ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Nous Montrons ◽  
Schröder Paths ◽  
International Audience ◽  
Finite Sum ◽  
Jacobi's Triple Product Identity

International audience We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. As a consequence we find a connection between his formula and Jacobi's triple product identity. We then give a combinatorial analog of Jacobi's triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple product identity is recovered by taking the limit. This can be stated in terms of some continued fractions called T-fractions, whose important property is the fact that they satisfy some functional equation. We show that this result permits to explain and generalize some Touchard-Riordan-like formulas appearing in enumerative problems. Nous donnons une preuve combinatoire d'une formule à la Touchard-Riordan due au premier auteur. En conséquence, nous faisons appara\^ıtre un lien entre cette formule et l'identité du produit triple de Jacobi. Nous donnons un analogue combinatoire à l'identité du produit triple en montrant qu'une somme finie peut être interprétée comme fonction génératrice de chemins de Schröder pondérés, de sorte que l'identité du produit triple s'obtient en passant à la limite. Ceci peut être énoncé en termes de fractions continues appelées T-fractions, dont la propriété importante est le fait qu'elle satisfont certaines équations fonctionnelles. Nous montrons que ce résultat permet d'expliquer et généraliser certaines formules à la Touchard-Riordan apparaissant dans des problèmes d'énumération.

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]).

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 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 Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

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