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

Kongsiriwong, Sarachai

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Description

Title

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

Author(s)

Kongsiriwong, Sarachai

Issue Date

2003

Doctoral Committee Chair(s)

Berndt, Bruce C.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

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.

Graduation Semester

2003

Type of Resource

text

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