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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The statistics of continued fractions
for polynomials over a finite field


Authors: Christian Friesen and Doug Hensley
Journal: Proc. Amer. Math. Soc. 124 (1996), 2661-2673
MSC (1991): Primary 11A55
MathSciNet review: 1328349
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Abstract: Given a finite field $F$ of order $q$ and polynomials $a,b\in F[X]$ of degrees $m<n$ respectively, there is the continued fraction representation $b/a=a_1+1/(a_2+1/(a_3+\dots +1/a_r))$. Let $CF(n,k,q)$ denote the number of such pairs for which $\deg b=n,\ \deg a<n,$ and for $1\le j\le r,$ $\deg a_j \le k$. We give both an exact recurrence relation, and an asymptotic analysis, for $CF(n,k,q)$. The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of $r$. Averaged over all $a$ and $b$ as above, this presents no difficulties. The average value of $r$ is $n(1-1/q)$, and there is full information about the distribution. When $b$ is fixed and only $a$ is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of $r$ that differs from this average by more than $O(\sqrt {n/q}).$


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Additional Information

Christian Friesen
Affiliation: Department of Mathematics, Ohio State University, Marion Campus, Marion, Ohio 43302
Email: friesen.4@osu.edu

Doug Hensley
Affiliation: Department of Mathematics, Texas A& M University, College Station, Texas 77843
Email: doug.hensley@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03394-1
PII: S 0002-9939(96)03394-1
Received by editor(s): August 20, 1994
Received by editor(s) in revised form: March 27, 1995
Communicated by: William W. Adams
Article copyright: © Copyright 1996 American Mathematical Society