Definition:Numerators and Denominators of Continued Fraction
- Not to be confused with Definition:Partial Numerator or Definition:Partial Denominator.
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Definition
Let $F$ be a field.
Let $n \in \N \cup \set \infty$ be an extended natural number.
Let $C = \sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $F$ of length $n$.
Definition 1: recursive definition
The sequence of numerators of $C$ is the sequence $\sequence {p_k}_{0 \mathop \le k \mathop \le n}$ that is recursively defined by:
- $p_k = \begin {cases}
a_0 & : k = 0 \\
a_1 a_0 + 1 & : k = 1 \\
a_k p_{k - 1} + p_{k - 2} & : k \ge 2
\end {cases}$
The sequence of denominators of $C$ is the sequence $\sequence {q_k}_{0 \mathop \le k \mathop \le n}$ that is recursively defined by:
- $q_k = \begin {cases}
1 & : k = 0 \\
a_1 & : k = 1 \\
a_k q_{k - 1} + q_{k - 2} & : k \ge 2
\end {cases}$
Definition 2: using matrix products
Let $k \ge 0$, and let the indexed matrix product:
- $\ds \prod_{i \mathop = 0}^k \begin {pmatrix}
a_i & 1 \\ 1 & 0
\end {pmatrix} = \begin {pmatrix}
x_{1 1}^{\paren k} & x_{1 2}^{\paren k} \\
x_{2 1}^{\paren k} & x_{2 2}^{\paren k}
\end {pmatrix}$ The $k$th numerator is $x_{1 1}^{\paren k}$ and the $k$th denominator is $x_{2 1}^{\paren k}$.
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