Definition:Numerators and Denominators of Continued Fraction

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

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:

$\displaystyle \prod_{i \mathop = 0}^k \begin {pmatrix} a_i & 1 \\ 1 & 0 \end {pmatrix} = \begin {pmatrix} x_{1 1}^{\paren k} & x_{1 2}^{\paren k} \\ x_{2 1}^{\paren k} & x_{2 2}^{\paren k} \end {pmatrix}$

The $k$th numerator is $x_{1 1}^{\paren k}$ and the $k$th denominator is $x_{2 1}^{\paren k}$.