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Generalized Continued Fraction


A generalized continued fraction is an expression of the form

 b_0+(a_1)/(b_1+(a_2)/(b_2+(a_3)/(b_3+...))),
(1)

where the partial numerators a_1,a_2,... and partial denominators b_0,b_1,b_2,... 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

 x=b_0+(a_1)/(b_1+)(a_2)/(b_2+)...
(2)

or

 x=b_0+K_(n=1)^infty(a_n)/(b_n).
(3)

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

Padé approximants provide another method of expanding functions, namely as a ratio of two power series. 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 1775). The Ramanujan continued fractions provide another fascinating class of continued fraction constants, and the Rogers-Ramanujan continued fraction is an example of a convergent generalized continued fraction function where a simple definition leads to quite intricate structure.

continued fractionvalueapproximateOEIS
K_(n=1)^(infty)1/k(I_1(2))/(I_0(2))0.697774...A052119
K_(n=1)^(infty)k/k(e-1)^(-1)0.581976...A073333
1+K_(n=1)^(infty)k/1sqrt(2/(epi))[erfc(2^(-1/2))]^(-1)1.525135...A111129
K_(n=1)^(infty)k/k(sqrt(e)-1)^(-1)1.541494...A113011

The value

 (A_n)/(B_n)=b_0+K_(k=1)^n(a_k)/(b_k)
(4)

is known as the nth convergent of the continued fraction.

A regular continued fraction representation (which is usually what is meant when the term "continued fraction" is used without qualification) of a number x is one for which the partial quotients are all unity (a_n=1), b_0 is an integer, and b_1, b_2, ... are positive integers (Rockett and Szüsz, 1992, p. 3).

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

 c_1+c_1c_2+c_1c_2c_3+...,
(5)

then it is equal to the continued fraction

 (c_1)/(1-(c_2)/(1+c_2-(c_3)/(1+c_3-...)))
(6)

(Borwein et al. 2004, p. 30).

To "round" a regular continued fraction, truncate the last term unless it is +/-1, in which case it should be added to the previous term (Gosper 1972, Item 101A). 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

 [-a,-b,-c,-d,...]=[-a-1,1,b-1,c,d,...].
(7)

A particularly beautiful identity involving the terms of the continued fraction is

 ([a_0,a_1,...,a_n])/([a_0,a_1,...,a_(n-1)])=([a_n,a_(n-1),...,a_1,a_0])/([a_n,a_(n-1),...,a_1]).
(8)

There are two possible representations for a finite simple fraction:

 [a_0,...,a_n]={[a_0,...,a_(n-1),a_n-1,1]   for a_n>1; [a_0,...,a_(n-2),a_(n-1)+1]   for a_n=1.
(9)

See also

Continued Fraction, Continued Fraction Constants, Convergent, Lehner Continued Fraction, Padé Approximant, Partial Denominator, Partial Numerator, Ramanujan Continued Fractions, Regular Continued Fraction, Rogers-Ramanujan Continued Fraction, Simple Continued Fraction

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Gosper, R. W. "Continued fractions query." math-fun@cs.arizona.edu posting, Dec. 27, 1996.Gosper, R. W. Item 101a in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 37-39, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/cf.html#item101a.Rockett, A. M. and Szüsz, P. Continued Fractions. New York: World Scientific, 1992.

Cite this as:

Weisstein, Eric W. "Generalized Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeneralizedContinuedFraction.html

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