§1.12 Continued Fractions

§1.12(i) Notation

The notation used throughout the DLMF for the continued fraction

1.12.1

is

1.12.2

§1.12(ii) Convergents

1.12.3,
1.12.4

is called the th approximant or convergent to . and are called the th (canonical) numerator and denominator respectively.

1.12.5
,
, ,
1.12.6

¶ Equivalence

Two continued fractions are equivalent if they have the same convergents.

is equivalent to if there is a sequence , ,
, such that

1.12.15,

and

1.12.16.

Formally,

1.12.17

when , .

1.12.19,

when , .

Define

1.12.20

Then

§1.12(iii) Existence of Convergents

A sequence in the extended complex plane, , can be a sequence of convergents of the continued fraction (1.12.3) iff

1.12.22
.

§1.12(iv) Contraction and Extension

A contraction of a continued fraction is a continued fraction whose convergents form a subsequence of the convergents of . Conversely, is called an extension of . If , , then is called the even part of . The even part of exists iff , , and up to equivalence is given by

1.12.23

If , , then is called the odd part of . The odd part of exists iff , , and up to equivalence is given by

1.12.24

§1.12(v) Convergence

A continued fraction converges if the convergents tend to a finite limit as .

¶ Pringsheim’s Theorem

The continued fraction converges when

1.12.25.

With these conditions the convergents satisfy and with .

¶ Van Vleck’s Theorem

Let the elements of the continued fraction satisfy

1.12.26,

where is an arbitrary small positive constant. Then the convergents satisfy

1.12.27,

and the even and odd parts of the continued fraction converge to finite values. The continued fraction converges iff, in addition,

1.12.28

In this case .

§1.12(vi) Applications

For analytical and numerical applications of continued fractions to special functions see §3.10.