\begin{definition}[Definition:Generalized Continued Fraction]
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 $((b_n)_{n\geq 0}, (a_n)_{n\geq 1})$ in $k$, called '''sequence of partial denominators''' and '''sequence of partial numerators''' respectively.
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\end{definition}