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§1.12 Continued Fractions

Contents
  1. §1.12(i) Notation
  2. §1.12(ii) Convergents
  3. §1.12(iii) Existence of Convergents
  4. §1.12(iv) Contraction and Extension
  5. §1.12(v) Convergence
  6. §1.12(vi) Applications

§1.12(i) Notation

The notation used throughout the DLMF for the continued fraction

1.12.1 b0+a1b1+a2b2+

is

1.12.2 b0+a1b1+a2b2+.

§1.12(ii) Convergents

1.12.3 C=b0+a1b1+a2b2+,
an0,
1.12.4 Cn=b0+a1b1+a2b2+anbn=AnBn.

Cn is called the nth approximant or convergent to C. An and Bn are called the nth (canonical) numerator and denominator respectively.

Recurrence Relations

1.12.5 Ak =bkAk1+akAk2,
Bk =bkBk1+akBk2,
k=1,2,3,,
1.12.6 A1 =1,
A0 =b0,
B1 =0,
B0 =1.

Determinant Formula

1.12.7 AnBn1BnAn1=(1)n1k=1nak,
n=0,1,2,.
1.12.8 CnCn1=(1)n1k=1nakBn1Bn,
n=1,2,3,,
1.12.9 Cn=b0+a1B0B1+(1)n1k=1nakBn1Bn.
1.12.10 an =An1BnAnBn1An1Bn2An2Bn1,
n=1,2,3,,
1.12.11 an =BnBn2Cn1CnCn1Cn2,
n=2,3,4,,
1.12.12 bn =AnBn2An2BnAn1Bn2An2Bn1,
n=1,2,3,,
1.12.13 bn =BnBn1CnCn2Cn1Cn2,
n=2,3,4,,
1.12.14 b0 =A0=C0,
b1 =B1,
a1 =A1A0B1.

Equivalence

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

b0+a1b1+a2b2+ is equivalent to b0+a1b1+a2b2+ if there is a sequence {dn}n=0, d0=1,
dn0, such that

1.12.15 an=dndn1an,
n=1,2,3,,

and

1.12.16 bn=dnbn,
n=0,1,2,.

Formally,

1.12.17 b0+a1b1+a2b2+a3b3+=b0+a1/b11+a2/(b1b2)1+a3/(b2b3)1+an/(bn1bn)1+=b0+1(1/a1)b1+1(a1/a2)b2+1(a2/(a1a3))b3+1(a1a3/(a2a4))b4+.

Series

1.12.18 p0+k=1np1p2pk=p0+p11p21+p2p31+p3pn1+pn,
n=0,1,2,,

when pk0, k=1,2,3,.

1.12.19 k=0nckxk=c0+c1x1(c2/c1)x1+(c2/c1)x(c3/c2)x1+(c3/c2)x(cn/cn1)x1+(cn/cn1)x,
n=0,1,2,,

when ck0, k=1,2,3,.

Fractional Transformations

Define

1.12.20 Cn(w)=b0+a1b1+a2b2+anbn+w.

Then

1.12.21 Cn(w) =An+An1wBn+Bn1w,
Cn(0) =Cn,
Cn() =Cn1=An1Bn1.

§1.12(iii) Existence of Convergents

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

1.12.22 C0 ,
Cn Cn1,
n=1,2,3,.

§1.12(iv) Contraction and Extension

A contraction of a continued fraction C is a continued fraction C whose convergents {Cn} form a subsequence of the convergents {Cn} of C. Conversely, C is called an extension of C. If Cn=C2n, n=0,1,2,, then C is called the even part of C. The even part of C exists iff b2k0, k=1,2,, and up to equivalence is given by

1.12.23 b0+a1b2a2+b1b2a2a3b4a3b4+b2(a4+b3b4)a4a5b2b6a5b6+b4(a6+b5b6)a6a7b4b8a7b8+b6(a8+b7b8).

If Cn=C2n+1, n=0,1,2,, then C is called the odd part of C. The odd part of C exists iff b2k+10, k=0,1,2,, and up to equivalence is given by

1.12.24 a1+b0b1b1a1a2b3/b1a2b3+b1(a3+b2b3)a3a4b1b5a4b5+b3(a5+b4b5)a5a6b3b7a6b7+b5(a7+b6b7).

§1.12(v) Convergence

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

Pringsheim’s Theorem

The continued fraction a1b1+a2b2+ converges when

1.12.25 |bn||an|+1,
n=1,2,3,.

With these conditions the convergents Cn satisfy |Cn|<1 and CnC with |C|1.

Van Vleck’s Theorem

Let the elements of the continued fraction 1b1+1b2+ satisfy

1.12.26 12π+δ<phbn<12πδ,
n=1,2,3,,

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

1.12.27 12π+δ<phCn<12πδ,
n=1,2,3,,

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

1.12.28 n=1|bn|=.

In this case |phC|12π.

§1.12(vi) Applications

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