Generalized Continued Fraction

A generalized continued fraction is an expression of the form

(1)

where the partial numerators and partial denominators may in general be integers, real numbers, complex numbers, or functions (Rockett and Szüsz 1992, p. 1). Generalized continued fractions may also be written in the forms

(2)

or

(3)

Note that letters other than are sometimes also used; for example, the documentation for ContinuedFractionK[f, g, i, imin, imax] in the Wolfram Language uses .

Padé approximants provide another method of expanding functions, namely as a ratio of two polynomials. The quotient-difference algorithm allows interconversion of continued fraction, power series, and rational function approximations.

A small sample of closed-form continued fraction constants is given in the following table (cf. Euler 1782). The Ramanujan continued fractions provide another fascinating class of continued fraction constants, and the Rogers-Ramanujan continued fraction is a generalized continued fraction function that converges for , where a simple definition leads to quite intricate structure.

continued fraction	value	approximate	OEIS
		0.697774...	A052119
		0.581976...	A073333
		1.525135...	A111129
		1.541494...	A113011

The value

(4)

is known as the th convergent of the continued fraction.

Calling a convergent does not assert that the infinite continued fraction converges. Ordinary convergence requires the sequence of convergents to approach a single limit. On the Riemann sphere, the limit may also be the point at infinity. Distinct subsequential limits imply divergence. The Stern-Stolz theorem gives a class with two distinct subsequential limits, while some Ramanujan continued fractions have three (Andrews et al. 2005; Bowman and Mc Laughlin 2007, Cor. 1).

A regular continued fraction representation (which is usually what is meant when the term "continued fraction" is used without qualification) of a number is one for which the partial numerators are all unity (), is an integer, and , , ... are positive integers (Rockett and Szüsz 1992, p. 3).

Euler showed that if a convergent series can be written in the form

(5)

then it is equal to the continued fraction

(6)

(Borwein et al. 2004, p. 30).

To "round" a regular continued fraction at a specified cutoff, truncate the remaining terms unless the first omitted term is , in which case add it to the last retained term (Gosper 1972, pp. 37-39). To take one over a simple continued fraction, add (or possibly delete) an initial 0 term. To negate, take the negative of all terms, optionally using the identity

(7)

For a finite continued fraction, the identity

(8)

holds whenever all the written ratios are defined. There are two possible representations for a finite simple continued fraction:

(9)