On Ramanujan's general theta function and a generalization of the Borweins' cubic theta functions

2015 ◽  
Vol 08 (01) ◽  
pp. 1550002
Author(s):  
Chandrashekar Adiga ◽  
Nasser Abdo Saeed Bulkhali
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Special Case

The Borwein brothers have introduced and studied three cubic theta functions. Many generalizations of these functions have been studied as well. In this paper, we introduce a new generalization of these functions and establish general formulas that are connecting our functions and Ramanujan's general theta function. Many identities found in the literature follow as a special case of our identities. We further derive general formulas for certain products of theta functions.

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

scholarly journals An alternate circular summation formula of theta functions and its applications

2012 ◽  
Vol 6 (1) ◽  
pp. 114-125 ◽  
Author(s):  
Jun-Ming Zhu
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Summation Formula ◽  
Theta Function Identities

We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulaes for (q; q)2n?.

scholarly journals Theta Functions and Abelian Varieties over Valuation Fields of Rank one I

1962 ◽  
Vol 20 ◽  
pp. 1-27 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Imaginary Part ◽  
Classical Theory ◽  
Theta Function ◽  
Abelian Varieties ◽  
Theta Functions ◽  
Positive Definite ◽  
Complex Matrix ◽  
Rank One ◽  
Symmetric Complex

We shall denote by the Z-module of integral vectors of dimension r, by T a symmetric complex matrix with positive definite imaginary part and by g the variable vector. If we put and the fundamental theta function is expressed in the form: as a series in q and u. Other theta functions in the classical theory are derived from the fundamental theta function by translating the origin and making sums and products, so these theta functions are also expressed in the form: as series of q and u. Moreover the coefficients in the relations of theta functions are also expressed in the form: as series in q.

Combinatorics of two second-order mock theta functions

2020 ◽  
pp. 1-11
Author(s):  
Hannah Burson
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Second Order ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Third Order ◽  
Combinatorial Interpretations ◽  
Special Cases ◽  
Order Mock Theta Function

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function [Formula: see text] and Ramanujan’s third-order mock theta function [Formula: see text].

Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms

2020 ◽  
pp. 1-10
Author(s):  
Amanda Folsom
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Roots Of Unity ◽  
Radial Limit ◽  
Special Values ◽  
Partial Theta Function ◽  
Finite Sums ◽  
Partial Theta Functions

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity.

SOME INVERSE RELATIONS AND THETA FUNCTION IDENTITIES

2012 ◽  
Vol 08 (08) ◽  
pp. 1977-2002 ◽  
Author(s):  
ZHI-GUO LIU
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Theta Function ◽  
Theta Functions ◽  
Fourier Series Expansion ◽  
Theta Function Identities ◽  
Trigonometric Identities

Two pairs of inverse relations for elliptic theta functions are established with the method of Fourier series expansion, which allow us to recover many classical results in theta functions. Many nontrivial new theta function identities are discovered. Some curious trigonometric identities are derived.

The number of contact primes of the canonical curve of genus p

1930 ◽  
Vol 26 (4) ◽  
pp. 453-457
Author(s):  
W. G. Welchman
Keyword(s):  
General Formula ◽  
Theta Functions ◽  
Cubic Surface ◽  
General Curve ◽  
Projective Model ◽  
Quartic Curve ◽  
Canonical Curve ◽  
Riemann Theta Functions ◽  
Sextic Curve ◽  
Special Case

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.

scholarly journals A decomposition theorem on differential polynomials of theta functions

1984 ◽  
Vol 96 ◽  
pp. 113-126 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Entire Function ◽  
Imaginary Part ◽  
Theta Function ◽  
Theta Functions ◽  
Decomposition Theorem ◽  
Positive Definite ◽  
Complex Variables ◽  
Differential Polynomials ◽  
Symmetric Complex ◽  
The Difference

Let τ = (τij) be a symmetric complex g = g matrix with the positive definite imaginary part. A theta function of level n means an entire function f(z) in g complex variables z = (zl …, zg) satisfying the difference relations:

DIFFERENTIAL EQUATIONS FOR CUBIC THETA FUNCTIONS

2011 ◽  
Vol 07 (07) ◽  
pp. 1945-1957 ◽  
Author(s):  
TIM HUBER
Keyword(s):  
Differential Equations ◽  
Eisenstein Series ◽  
Theta Function ◽  
Modular Group ◽  
Short Proof ◽  
Nonlinear Differential Equations ◽  
Theta Functions ◽  
Coupled Systems ◽  
Function Identity ◽  
Theta Function Identity

We show that the cubic theta functions satisfy two distinct coupled systems of nonlinear differential equations. The resulting relations are analogous to Ramanujan's differential equations for Eisenstein series on the full modular group. We deduce the cubic analogs presented here from trigonometric series identities arising in Ramanujan's original paper on Eisenstein series. Several consequences of these differential equations are established, including a short proof of a famous cubic theta function identity derived by J. M. Borwein and P. B. Borwein.

THE COEFFICIENTS OF THE ω(q) MOCK THETA FUNCTION

2008 ◽  
Vol 04 (06) ◽  
pp. 1027-1042 ◽  
Author(s):  
SHARON ANNE GARTHWAITE
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Theta Functions ◽  
Exact Formula ◽  
Poincaré Series ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Poincare Series ◽  
Vector Valued

In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of these functions. Recently, Zwegers proved a relationship between mock theta functions and vector-valued modular forms, and Bringmann and Ono used the theory of Maass forms and Poincaré series to prove a conjecture of Andrews, yielding an exact formula for the coefficients of the f(q) mock theta function. Here we build upon these results, using the theory of vector-valued modular forms and Poincaré series to prove an exact formula for the coefficients of the ω(q) mock theta function.

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