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 real or complex values.

A generalized continued fraction is an expression of the form

where the an (n > 0) are the partial numerators, the bn 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:

x_0 = \frac{A_0}{B_0} = b_0, \qquad x_1 = \frac{A_1}{B_1} = \frac{b_1b_0+a_1}{b_1},\qquad x_2 = \frac{A_2}{B_2} = \frac{b_2(b_1b_0+a_1) + a_2b_0}{b_2b_1 + a_2},\qquad\cdots\,

and in general

A_n = b_n A_{n-1} + a_n A_{n-2}, \qquad B_n = b_n B_{n-1} + a_n B_{n-2}, \,

where An is the numerator and Bn is the denominator, called continuants, of the nth convergent.

If the sequence of convergents {xn} 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 Bn.

Read more about Generalized Continued Fraction:  History of Continued Fractions, Notation, Some Elementary Considerations, Linear Fractional Transformations, Continued Fractions and Series, Higher Dimensions

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