Infinite Series Identities in the Theory of Elliptic Functions and Q-Series

We prove several infinite series identities. In Chapter 2, We extend C. L. Siegel's method of proving the Dedekind-eta function transformation by integrating some selected functions over a positively oriented polygon, generalizing Siegel's integration over a parallelogram. As consequences, we obtain a generalization of the Dedekind-eta function transformation and generalizations of other transformation formulas. In Chapter 3, we adapt B. C. Berndt and A. Zaharescu's method to establish a multi-variable theta product identity of a function of k + 1 complex variables. In Chapter 4, we give a simple new proof of the classical theta-function inversion formula. In Chapter 5, we give two general methods for proving q-series-product identities. The first method uses basic properties of roots of unity. The second method generalizes S. Bhragava's argument proving the quintuple product identity. By comparing the results from the two methods, we obtain new identities. Using these identities, we can derive certain modular equations. In Chapter 6, we evaluate certain infinite series involving hyperbolic functions by using the cubic theory of elliptic functions.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Kongsiriwong, Sarachai

Contributors dc:contributor

• Berndt, Bruce C.

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3111562

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86827