Definition:Generalized Continued Fraction
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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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continued fraction
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continued fraction
- Weisstein, Eric W. "Generalized Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeneralizedContinuedFraction.html