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

hypergeometric functions;continued fractions;gamma function;basic hypergeometric functions;q-analogue;q-series;Ramanujan;"Ramanujans notebooks"]

Reuter, Victoria

关键词: hypergeometric functions;    continued fractions;    gamma function;    basic hypergeometric functions;    q-analogue;    q-series;    Ramanujan;    "Ramanujans notebooks"];   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/72779/Victoria_Reuter.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Some of the most interesting of Ramanujan's continued fraction identities are those involvingratios of Gamma functions in Chapter 12 of his second notebook. This thesis developsa 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'sidentities. We next prove Ramanujan's Entries 36 and 39. Finally, we rework the methodfor use with basic hypergeometric functions and use it to find q-analogues of the earlier newresults.