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https://hdl.handle.net/2142/86899
Description
Title
Product Identities for Theta Functions
Author(s)
Cao, Zhu
Issue Date
2008
Doctoral Committee Chair(s)
Hildebrand, A.J.
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
In this thesis, firstly we derive a general method for establishing q-series 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 Gollnitz-Gordon functions found by S.-S. Huang in [34], and identities involving septic Rogers-Ramanujan 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 Blecksmith-Brillhart-Gerst 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.
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