Generalizations of Certain Results on Continued Fraction

In this thesis we study generalizations of the Rogers-Ramanujan continued fraction. The Rogers-Ramanujan continued fraction arises from a three-term q-difference equation. We consider (m + 1)-term q-difference equations and also a generalization of the continued fraction algorithm called a G-continued fraction. We obtain a general expansion of the quotient of two contiguous basic hypergeometric function in arbitrarily many variables as a G-continued fraction. A careful interpretation of convergence is given for different cases of this expansion. When a full vector space of solutions of a q-difference equation is known, we use the theorem of Zahar which extends a theorem of Pincherle. When this is not the case, we apply the theory on infinite system of equations to the G-continued fraction in order to obtain convergence. Also, an explicit formula for the approximants of a G-continued fraction is given. An application of this formula is used to obtain a combinatorial interpretation of a G-continued fraction extension of the Rogers-Ramanujan continued fraction. A combinatorial interpretation of the coefficients of the q-difference equation for a very well-poised basic hypergeometric series studied by A. Selberg is derived. Finally, the arithmetic properties of a generalization of the Rogers-Ramanujan continued fraction are considered.

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

• Choi, Geumlan

Contributors dc:contributor

• Douglas Bowman

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3017047

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86775