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Title:  Product Identities for Theta Functions 
Author(s):  Cao, Zhu 
Doctoral Committee Chair(s):  Hildebrand, A.J. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  In this thesis, firstly we derive a general method for establishing qseries identities. Using this method, we can show that if Zn can be taken as the disjoint union of a lattice generated by n linearly independent vectors in Z n and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. Many known identities, including M. D. Hirschhorn's generalization of the quintuple product identity, several modular relations for the GollnitzGordon functions found by S.S. Huang in [34], and identities involving septic RogersRamanujan functions obtained by H. Hahn in [29], are shown to be special cases of this general formula. We also obtain a generalization of the septuple product identity. Several entries in Ramanujan's notebooks as well as new identities are proved as applications, including an analogue of Winquist's identity and a new representation of (q; q&parr0;8infinity . We also give several generalized forms of Schroter's formula. A general theorem by W. Chu and Q. Yan [17] and the BlecksmithBrillhartGerst theorem in [11] are both special cases of our generalized Schroter formula. In Chapter 3, we give a proof of a generalized form of a reciprocity theorem in Ramanujan's lost notebook. Also we generalize the results obtained by S.Y. Kang in [38]. Then several new reciprocity theorems and their applications are presented. In Chapter 4, we prove the Jacobi triple product identity, the quintuple product identity, and the septuple product identity using properties of cubic and fifth roots of unity. Chapter 5 is devoted to new proofs of Winquist's identity and the septuple product identity. Lastly, in Chapter 6, we give a generalization of Heine's transformation and some applications. 
Issue Date:  2008 
Type:  Text 
Language:  English 
Description:  110 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2008. 
URI:  http://hdl.handle.net/2142/86899 
Other Identifier(s):  (MiAaPQ)AAI3314738 
Date Available in IDEALS:  20150928 
Date Deposited:  2008 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois