Definition:Numerators and Denominators of Continued Fraction

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Not to be confused with Definition:Partial Numerator or Definition:Partial Denominator.

Definition

Let $F$ be a field.

Let $n\in \N\cup \{\infty\}$ be an extended natural number.

Let $C = \left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $F$ of length $n$.


Definition 1: recursive definition

The sequence of numerators of $C$ is the sequence $(p_k)_{0 \leq k \leq n}$ that is recursively defined by:

$p_k = \begin{cases} a_0 & : k = 0 \\ a_0 a_1 + 1 & : k = 1 \\ a_k p_{k - 1} + p_{k - 2} & : k \geq 0 \end{cases}$

The sequence of denominators of $C$ is the sequence $(q_k)_{0 \leq k \leq n}$ that is recursively defined by:

$q_k = \begin{cases} 1 & : k = 0 \\ a_1 & : k = 1 \\ a_k q_{k - 1} + q_{k - 2} & : k \geq 0 \end{cases}$


Definition 2: using matrix products

Let $k \geq 0$, and let the indexed matrix product:

$\displaystyle \prod_{i = 0}^k\begin{pmatrix} a_i & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} x_{11}^{(k)} & x_{12}^{(k)} \\ x_{21}^{(k)} & x_{22}^{(k)} \end{pmatrix}$

The $k$th numerator is $x_{11}^{(k)}$ and the $k$th denominator is $x_{21}^{(k)}$.


Also see