Continued fraction

An expression of the form

(1)

where

(2)

and

(3)

are finite or infinite sequences of complex numbers. Instead of the expression (1) one also uses the notation

The continued fraction of the sequence (2) is defined as the expression

For every continued fraction (1) the recurrence equations

with the initial conditions

determine two sequences and of complex numbers. As a rule, it is assumed that the sequences (2) and (3) are such that for all , . The fraction is called the -th convergent of the continued fraction (1). Here

moreover,

It is convenient to denote the -th convergent of the continued fraction of the sequence (2) by

These convergents satisfy the following equalities:

If and the sequence of convergents of (1) converges to some limit , then the continued fraction (1) is called convergent and the number is its value. If , that is, the continued fraction is finite, then its value is defined as the last of its convergents.

If all terms of the sequences (2) and (3), except possibly , are positive real numbers, and if is real, then the sequence of convergents of even order of (1) increases, and the sequence of convergents of odd order decreases. Here a convergent of even order is less than the corresponding convergent of odd order (see [5]).

If is the sequence of complex numbers for which

then the expression (1) is called an expansion of the number in a continued fraction. Not every continued fraction converges, and the value of a continued fraction is not always equal to the number from which it is expanded. There are a number of criteria for the convergence of continued fractions (see, for example, [3] and [5]):

1) Suppose that , that all terms of the sequences (2) and (3) are real numbers, and that for all natural numbers from some term onwards. If for such , then the continued fraction (1) converges.

2) Suppose that and that all terms of the sequence (2) beginning with are positive. Then the continued fraction of the sequence (2) converges if and only if the series diverges (Seidel's theorem).

The continued fraction of a sequence (2) is called regular if all its terms (except possibly ) are natural numbers, is an integer and for . For every real number there exists a unique regular continued fraction with value . This fraction is finite if and only if is rational (see [1], [2], [4]). An algorithm for the expansion of a real number in a regular continued fraction is defined by the following relations

(4)

where denotes the integral part of .

The numbers and defined by (4) are called, respectively, the complete and incomplete quotients of order of the expansion of in a continued fraction.

Around 1768 J. Lambert found the expansion of in a continued fraction:

Under the assumption that this continued fraction converges, A. Legendre proved that its value for rational values of is irrational. It should be mentioned that in this way he proved the irrationality of the number (see [7]).

L. Euler found in 1737 that

A real number is an irrational root of a polynomial of degree 2 with integer coefficients if and only of the incomplete quotients of the expansion of in a continued fraction from some term onwards are repeated periodically (the Euler–Lagrange theorem, see [1] and [4]). At present (1984) expansions in regular continued fractions of algebraic numbers of degree 3 and higher are not known. The assertion that the incomplete quotients of the expansion of in a continued fraction are bounded has not been proved.

Regular continued fractions are a very convenient tool for the approximation of real numbers by rational numbers. The following propositions hold:

1) If and are neighbouring convergents of the expansion of a number in a regular continued fraction, then

and

where in the latter case equality holds only when .

2) For two neighbouring convergents of the expansion of a number in a regular continued fraction, at least one of them satisfies the inequality:

3) If and are integers, , if is a real number, and if

then is a convergent of the expansion of in a regular continued fraction.

4) If is a convergent of the expansion of a number into a regular continued fraction, then for any integers and it follows from , and

that (theorem on the best approximation).

The first twenty-five incomplete quotients of the expansion of the number in a regular continued fraction are: 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1.

The first five convergents of the expansion of in a regular continued fraction are:

Therefore,

There exist several generalizations of continued fractions (see, e.g., [9]).

References

[1]	A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)
[2]	B.A. Venkov, "Elementary number theory" , Wolters-Noordhoff (1970) (Translated from Russian)
[3]	A.A. Markov, "Selected works" , Moscow-Leningrad (1948) (In Russian)
[4]	A.Ya. [A.Ya. Khinchin] Khintchine, "Kettenbrüche" , Teubner (1956) (Translated from Russian)
[5]	A.N. Khovanskii, "Application of continued fractions and their generalizations to problems in approximation theory" , Moscow (1966) (In Russian)
[6]	, The history of mathematics from Antiquity to the beginning of the XIX-th century , 3 , Moscow (1972) (In Russian)
[7]	, Ueber die Kwadratur des Kreises (1936)
[8]	O. Perron, "Die Lehre von den Kettenbrüchen" , 1–2 , Teubner (1954–1957)
[9]	G. Szekeres, "Multidimensional continued fractions" Ann. Univ. Sci. Sec. Math. , 13 (1970) pp. 113–140
[10]	W.B. Jones, W.J. Thron, "Continued fractions, analytic theory and applications" , Addison-Wesley (1980)

Comments

A classical reference on convergence is [a1]. Together with [10], recent references are [a2]–[a4]. Some generalizations can be found in [a5]–[a7]. Except for [a1], [a8] all references contain extensive lists of references on recent developments and applications in Padé approximation; moment problems (cf. Moment problem); orthogonal polynomials; number theory; and the metrical theory of continued fractions (see also Metric theory of numbers).

References

[a1]	H.S. Wall, "Analytic theory of continued fractions" , Chelsea (1973)
[a2]	W.B. Jones (ed.) W.J. Thron (ed.) E.H. Waadeland (ed.) , Analytic theory of continued fractions , Lect. notes in math. , 932 , Springer (1982)
[a3]	W.J. Thron (ed.) , Analytic theory of continued fractions II , Lect. notes in math. , 1199 , Springer (1986)
[a4]	C. Kraaikamp, "The distribution of some sequences connected with the nearest integer continued fraction" Indag. Math. , 49 (1987) pp. 177–191
[a5]	A.J. Brentjes, "Multidimensional continued fractions" , CWI , Amsterdam (1981)
[a6]	V.Ya. Skorobogatko, "The theory of convergent continued fractions and its applications in numerical mathematics" , Moscow (1983) (In Russian)
[a7]	D.I. Bodnap, "Convergent continued fractions" , Kiev (1986) (In Russian)
[a8]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1959)
[a9]	P. Henrici, "Applied and computational complex analysis" , 2 , Wiley (1977)