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Berstein's Families

The following is a list of families of continued fractions obtained by Berstein and published in [1]. The partial quotients in these families are predictable, and the functions for them are given in his paper, but to save space only the family and its primitive period length are given. Using the notation and variables of Levesque and Rhin in [3] (Berstein's families are also listed here), M will denote a family of continued fractions and A, B, and C will be as they are used in [3]. The list below also appears in the same order and format as it does in [3].

Let A = 2a + 1, where a and k are positive integers. The families associated with A are the following:
\begin{align*}M & = (A^k + a)^2 + A, \qquad \ell = 6k\\
M & = (A^k - a)^2 + A,...
...d \ell = 4k + 2\\
M & = (A^k - a - 1)^2 - A, \qquad \ell = 8k - 4
\end{align*}
Let B = 2d(2a - 1) such that a and d are positive integers with ad > 1. The families associated with B are the following:
\begin{align*}M & = (B^{k+1} + B - 1)^2 + 4B, \qquad \ell = 5k - 1\\
M & = (B^...
...l = 6k - 5\\
M & = (B^{k+1} - B - 1)^2 - 4B, \qquad \ell = 4k + 2
\end{align*}
For the remainder of this section, let a and k be natural numbers. Berstein obtained the following families, where C = 2d(2a-1) + 1:
\begin{align*}M & = (C^{6k-2} + C - 1)^2 + 4C, \qquad \ell = 22k - 12\\
M & = ...
...k - 10\\
M & = (C^{6k + 1} - C - 1)^2 - 4C, \qquad \ell = 24k - 2
\end{align*}
Berstein obtained 12 more families, but their form and primitive period length are not as easily stated as the families above.


next up previous
Next: Levesque and Rhin's Families Up: Families of Continued Fractions Previous: Families of Continued Fractions
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2000-01-06