Generalized continued fraction

In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.

A generalized continued fraction is an expression of the form

${\displaystyle x=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots \,}}}}}}}}}$

where the a_n (n > 0) are the partial numerators, the b_n are the partial denominators, and the leading term b₀ is called the integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:

${\displaystyle {\begin{aligned}x_{0}&={\frac {A_{0}}{B_{0}}}=b_{0},\\[4px]x_{1}&={\frac {A_{1}}{B_{1}}}={\frac {b_{1}b_{0}+a_{1}}{b_{1}}},\\[4px]x_{2}&={\frac {A_{2}}{B_{2}}}={\frac {b_{2}(b_{1}b_{0}+a_{1})+a_{2}b_{0}}{b_{2}b_{1}+a_{2}}},\ \dots \end{aligned}}}$

where A_n is the numerator and B_n is the denominator, called continuants,^[1][2] of the nth convergent. They are given by the recursion ^[3]

${\displaystyle {\begin{aligned}A_{n}&=b_{n}A_{n-1}+a_{n}A_{n-2},\\B_{n}&=b_{n}B_{n-1}+a_{n}B_{n-2}\qquad {\text{for }}n\geq 1\end{aligned}}}$

with initial values

${\displaystyle {\begin{aligned}A_{-1}&=1,&A_{0}&=b_{0},\\B_{-1}&=0,&B_{0}&=1.\end{aligned}}}$

If the sequence of convergents {x_n} approaches a limit the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators B_n.