On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

Authors:
Douglas Bowman and James Mc Laughlin

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3325-3347

MSC (2000):
Primary 11A55; Secondary 40A15

DOI:
https://doi.org/10.1090/S0002-9947-03-03390-7

Published electronically:
December 15, 2003

MathSciNet review:
2052952

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction.

Let the continued fraction expansion of any irrational number be denoted by and let the -th convergent of this continued fraction expansion be denoted by . Let

where . Let . It is shown that if , then the Rogers-Ramanujan continued fraction diverges at . is an uncountable set of measure zero. It is also shown that there is an uncountable set of points such that if , then does not converge generally.

It is further shown that does not converge generally for . However we show that does converge generally if is a primitive -th root of unity, for some . Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

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Additional Information

**Douglas Bowman**

Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115

Email:
bowman@math.niu.edu

**James Mc Laughlin**

Affiliation:
Department of Mathematics, Trinity College, 300 Summit Street, Hartford, Connecticut 06106-3100

Email:
james.mclaughlin@trincoll.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03390-7

Keywords:
Continued fractions,
Rogers-Ramanujan

Received by editor(s):
January 17, 2003

Received by editor(s) in revised form:
April 15, 2003

Published electronically:
December 15, 2003

Additional Notes:
The second author’s research supported in part by a Trjitzinsky Fellowship.

Article copyright:
© Copyright 2003
American Mathematical Society