scholarly journals A Note on Certain Modular Equations about Infinite Products of Ramanujan

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Hong-Cun Zhai
Keyword(s):  
Theta Functions ◽  
Modular Equations ◽  
Infinite Products

Ramanujan proposed additive formulae of theta functions that are related to modular equations about infinite products. Employing these formulaes, we derived some identities on infinite products. In the same spirit, we also could present elementary and simple proofs of certain Ramanujan's modular equations on infinite products.

scholarly journals Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

2010 ◽  
Vol 47 (04) ◽  
pp. 1023-1033 ◽  
Author(s):  
A. Kuznetsov
Keyword(s):  
Lévy Processes ◽  
Series Representation ◽  
Theta Functions ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transcendental Equation ◽  
Simple Expression ◽  
Lévy Measure ◽  
Infinite Products ◽  
Wide Range

In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

VANISHING COEFFICIENTS IN QUOTIENTS OF THETA FUNCTIONS OF MODULUS FIVE

2020 ◽  
Vol 102 (3) ◽  
pp. 387-398
Author(s):  
SHANE CHERN ◽  
DAZHAO TANG
Keyword(s):  
Infinite Product ◽  
Theta Functions ◽  
Infinite Products ◽  
General Family

Following recent investigations of vanishing coefficients in infinite products, we show that such instances are very rare when the infinite product is among a family of theta-quotients of modulus five. We also prove that a general family of products of theta functions of modulus five can always be effectively 5-dissected.

scholarly journals On a pair of Ramanujan’s modular equations and P - Q theta functions of level 35

2019 ◽  
Vol 43 (1) ◽  
pp. 63-80
Author(s):  
Kaliyur Ranganna VASUKI ◽  
Anusha THIPPESHA
Keyword(s):  
Theta Functions ◽  
Modular Equations

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.

Sign in / Sign up

Export Citation Format

Share Document