ON THE SCHRÖTER FORMULA FOR THETA FUNCTIONS

2009 ◽  
Vol 05 (08) ◽  
pp. 1477-1488 ◽  
Author(s):  
ZHI-GUO LIU ◽  
XIAO-MEI YANG
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Modular Equations ◽  
Trigonometric Identity ◽  
Product Identity ◽  
Function Identity ◽  
Special Cases ◽  
Theta Function Identity ◽  
Addition Formulas

The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

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.

scholarly journals A universal identity for theta functions of degree eight and applications

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Zhi-Guo Liu
Keyword(s):  
Theta Function ◽  
Natural Extension ◽  
Theta Functions ◽  
Common Source ◽  
Modular Functions ◽  
Sigma Function ◽  
Function Identity ◽  
International Audience ◽  
Theta Function Identity ◽  
Theta Function Identities

International audience Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

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].

scholarly journals SHIFTED AND SHIFTLESS PARTITION IDENTITIES II

2007 ◽  
Vol 03 (01) ◽  
pp. 43-84 ◽  
Author(s):  
FRANK G. GARVAN ◽  
HAMZA YESILYURT
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Computer Search ◽  
Modular Functions ◽  
Partition Identities ◽  
Partition Identity ◽  
Function Identity ◽  
Theta Function Identity ◽  
Fixed Positive Integer ◽  
Positive Integers

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

Combinatorial proof of a partial theta function identity of Warnaar

2016 ◽  
Vol 12 (06) ◽  
pp. 1475-1482 ◽  
Author(s):  
Kathy Q. Ji ◽  
Byungchan Kim ◽  
Jang Soo Kim
Keyword(s):  
Number Theory ◽  
Theta Function ◽  
Combinatorial Proof ◽  
Asymptotic Behaviors ◽  
Unimodal Sequences ◽  
Partial Theta Function ◽  
Function Identity ◽  
Theta Function Identity

By constructing a sign-reversing involution, we prove Warnaar’s identity involving a partial theta function, which plays many important roles in the study of asymptotic behaviors and quantum modularities in number theory. We also obtain an Euler-like theorem for a certain kind of unimodal sequences from Warnaar’s identity.

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