A twisted generalization of the classical Dedekind sum

2020 ◽  
pp. 1-16
Author(s):  
Brad Isaacson
Keyword(s):  
Number Theory ◽  
Theta Function ◽  
Modular Forms ◽  
Character Sums ◽  
Bernoulli Numbers ◽  
Dedekind Sum ◽  
Basic Facts ◽  
Theta Function Identities

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.

scholarly journals FINITE TRIGONOMETRIC CHARACTER SUMS VIA DISCRETE FOURIER ANALYSIS

2010 ◽  
Vol 06 (01) ◽  
pp. 51-67 ◽  
Author(s):  
MATTHIAS BECK ◽  
MARY HALLORAN
Keyword(s):  
Fourier Analysis ◽  
Theta Function ◽  
Character Sums ◽  
Contour Integration ◽  
Trigonometric Functions ◽  
Discrete Fourier Analysis ◽  
Complex Contour ◽  
Elementary Methods ◽  
Finite Sums ◽  
Theta Function Identities

We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were first systematically studied by Berndt and Zaharescu where their proofs involved complex contour integration. We show how to prove most of Berndt–Zaharescu's and some new identities by elementary methods of discrete Fourier analysis.

TWO THETA-FUNCTION IDENTITIES OF LEVEL 10

2020 ◽  
Vol 9 (7) ◽  
pp. 4929-4936
Author(s):  
D. Anu Radha ◽  
B. R. Srivatsa Kumar ◽  
S. Udupa
Keyword(s):  
Theta Function ◽  
Theta Function Identities

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.

TRANSFORMATION FORMULAS FOR THE NUMBER OF REPRESENTATIONS OF BY LINEAR COMBINATIONS OF FOUR TRIANGULAR NUMBERS

2020 ◽  
Vol 102 (1) ◽  
pp. 39-49
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Theta Function ◽  
Linear Combinations ◽  
Triangular Numbers ◽  
Transformation Formulas ◽  
Positive Integers ◽  
Theta Function Identities

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$, let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.

New truncated theorems for three classical theta function identities

2022 ◽  
Vol 101 ◽  
pp. 103470
Author(s):  
Ernest X.W. Xia ◽  
Ae Ja Yee ◽  
Xiang Zhao
Keyword(s):  
Theta Function ◽  
Theta Function Identities

A Pair of Theta Function Identities (L. Carlitz)

SIAM Review ◽  
1974 ◽  
Vol 16 (4) ◽  
pp. 553-555
Author(s):  
G. E. Andrews
Keyword(s):  
Theta Function ◽  
Theta Function Identities

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 function identities from optical neural network transformations

1993 ◽  
Vol 16 (4) ◽  
pp. 805-810
Author(s):  
E. Elizalde ◽  
A. Romeo
Keyword(s):  
Neural Network ◽  
Theta Function ◽  
Heisenberg Model ◽  
Symmetric Functions ◽  
New Approach ◽  
Jacobi Theta Function ◽  
Cylindrically Symmetric ◽  
Optical Neural Network ◽  
Theta Function Identities

We take a new approach to the generation of Jacobi theta function identities. It is complementary to the procedure which makes use of the evaluation of Parseval-like identities for elementary cylindrically-symmetric functions on computer holograms. Our method is more simple and explicit than this one, which was an outcome of the construction of neurocomputer architectures through the Heisenberg model.

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.

Sign in / Sign up

Export Citation Format

Share Document