scholarly journals A three-term theta function identity and its applications

2005 ◽  
Vol 195 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Zhi-Guo Liu
Keyword(s):  
Theta Function ◽  
Function Identity ◽  
Theta Function Identity

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.

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.

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.

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.

A mock theta function identity related to the partition rank modulo 3 and 9

2020 ◽  
pp. 1-17
Author(s):  
Song Heng Chan ◽  
Nankun Hong ◽  
Jerry ◽  
Jeremy Lovejoy
Keyword(s):  
Generating Function ◽  
Theta Function ◽  
Generating Functions ◽  
Mock Theta Function ◽  
Function Identity ◽  
Theta Function Identity ◽  
Cranks Of Partitions

We prove a new mock theta function identity related to the partition rank modulo 3 and 9. As a consequence, we obtain the [Formula: see text]-dissection of the rank generating function modulo [Formula: see text]. We also evaluate all of the components of the rank–crank differences modulo [Formula: see text]. These are analogous to conjectures of Lewis [The generating functions of the rank and crank modulo 8, Ramanujan J. 18 (2009) 121–146] on rank–crank differences modulo [Formula: see text], first proved by Mortenson [On ranks and cranks of partitions modulo 4 and 8, J. Combin. Theory Ser. A 161 (2019) 51–80].

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