Hypergeometric functions, continued fractions for products of gamma functions, and q-analogues

Reuter, Victoria

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https://hdl.handle.net/2142/72779 Copy

Description

Title

Hypergeometric functions, continued fractions for products of gamma functions, and q-analogues

Author(s)

Reuter, Victoria

Issue Date

2015-01-21

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

Berndt, Bruce C.

Doctoral Committee Chair(s)

Reznick, Bruce

Committee Member(s)

• Berndt, Bruce C.
• Hildebrand, A.J.
• Stolarsky, Kenneth B.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• hypergeometric functions
• continued fractions
• gamma function
• basic hypergeometric functions
• q-analogue
• q-series
• Ramanujan
• Ramanujan's notebooks

Abstract

Some of the most interesting of Ramanujan's continued fraction identities are those involving ratios of Gamma functions in Chapter 12 of his second notebook. This thesis develops a method for deriving such identities, using hypergeometric functions as the main tool. We begin by deriving a continued fraction identity, use it to prove Ramanujan's Entry 34, and then use the method to obtain new identities and relate them to two of Ramanujan's identities. We next prove Ramanujan's Entries 36 and 39. Finally, we rework the method for use with basic hypergeometric functions and use it to find q-analogues of the earlier new results.

Graduation Semester

2014-12

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http://hdl.handle.net/2142/72779

Copyright and License Information

Copyright 2014 Victoria Jane Reuter

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