Definition:Generalized Continued Fraction

From ProofWiki
Jump to navigation Jump to search

Definition

Let $k$ be a field.


Informally, a generalized continued fraction in $k$ is an expression of the form:

$b_0 + \cfrac {a_1} {b_1 + \cfrac {a_2} {b_2 + \cfrac {a_3} {\ddots \cfrac {} {b_{n - 1} + \cfrac {a_n} {b_n + \cfrac {a_{n + 1} } \ddots } } } } }$


Formally, a generalized continued fraction in $k$ is a pair of sequences $\tuple {\sequence {b_n}_{n \mathop \ge 0}, \sequence {a_n}_{n \mathop \ge 1} }$ in $k$, called the sequence of partial denominators and sequence of partial numerators respectively.


Also known as

A generalized continued fraction is also known as a general continued fraction.

Some sources refer to this merely as a continued fraction, without requiring that the $a_k$ coefficients be $1$.


Also see


Sources