Generalizations of Certain Results on Continued Fraction
Choi, Geumlan
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https://hdl.handle.net/2142/86775
Description
Title
Generalizations of Certain Results on Continued Fraction
Author(s)
Choi, Geumlan
Issue Date
2001
Doctoral Committee Chair(s)
Douglas Bowman
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
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.
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