Identities involving theta functions and analogues of theta functions

Xu, Ping

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https://hdl.handle.net/2142/45365 Copy

Description

Title

Identities involving theta functions and analogues of theta functions

Author(s)

Xu, Ping

Issue Date

2013-08-22T16:37:57Z

Director of Research (if dissertation) or Advisor (if thesis)

Berndt, Bruce C.

Doctoral Committee Chair(s)

Ahlgren, Scott

Committee Member(s)

• Berndt, Bruce C.
• Stolarsky, Kenneth B.
• Zaharescu, Alexandru

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Circular summation formula
• elementary proof
• two-dimensional lattice sums
• Poisson equation
• theta functions
• analogue of theta functions
• analogue of Gauss sums

Abstract

My dissertation is mainly about various identities involving theta functions and analogues of theta functions. In Chapter 1, we give a completely elementary proof of Ramanujan's circular summation formula of theta functions and its generalizations given by S. H. Chan and Z. -G. Liu, and J. M. Zhu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given. In Chapter 2, we analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many nice explicit examples are given. In Chapter 3, we study one page in Ramanujan's lost notebook that is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this chapter, we prove all the claims made by Ramanujan about this integral.

Graduation Semester

2013-08

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http://hdl.handle.net/2142/45365

Copyright and License Information

Copyright 2013 Ping Xu

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