### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Drew Sills, Rutgers University, asills {at} math [dot] rutgers [dot] edu
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Continued Fractions, and Generalizations, with Many Limits

Speaker: James McLaughlin, West Chester University

Date: Thursday, November 2, 2006 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Consider the following two examples of infinite processes:

(1) Let the sequence $\{x_n\}_{n \geq 0}$ be defined by $x_0 =\frac{4}{3}, \hspace{40pt} x_{n+1} = \frac{4}{3}-\frac{1}{x_n},\hspace{40pt} n\geq 1.$ What can be said about the sequence $\{x_n\}$?

(2) Let $\alpha$ and $\beta$ be points on the unit circle, let $|q|<1$, and define the continued fraction $G(q,\alpha, \beta):= \frac{- \alpha\beta}{\alpha+\beta+q} \+ \frac{-\alpha\beta}{\alpha+\beta+q^{2}} \+ \cds \+\frac{-\alpha\beta}{\alpha+\beta+q^{n}} \+ \cds .$

What can be said about the convergence or divergence of $G(q,\alpha, \beta)$? What difference, if any, does it make if $\alpha$ and $\beta$ are roots of unity?

It turns out that both the sequence and the continued fraction diverge, but diverge in quite interesting ways.

There are several infinite processes (matrix products, continued fractions, $(r,s)$-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way.

This talk will present a survey of results in the area, concentrating on recent results by Douglas Bowman and the speaker.