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

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Reuter, Victoria

Contributors dc:contributor

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

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

Statement dc:rights

• Copyright 2014 Victoria Jane Reuter

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/72779