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§1.12 Continued Fractions

Contents

§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 =bkAk-1+akAk-2,
Bk =bkBk-1+akBk-2,
k=1,2,3,,
1.12.6 A-1 =1,
A0 =b0,
B-1 =0,
B0 =1.

Determinant Formula

1.12.7 AnBn-1-BnAn-1=(-1)n-1k=1nak,
n=0,1,2,.
1.12.8 Cn-Cn-1=(-1)n-1k=1nakBn-1Bn,
n=1,2,3,,
1.12.9 Cn=b0+a1B0B1-+(-1)n-1k=1nakBn-1Bn.
1.12.10 an =An-1Bn-AnBn-1An-1Bn-2-An-2Bn-1,
n=1,2,3,,
1.12.11 an =BnBn-2Cn-1-CnCn-1-Cn-2,
n=2,3,4,,
1.12.12 bn =AnBn-2-An-2BnAn-1Bn-2-An-2Bn-1,
n=1,2,3,,
1.12.13 bn =BnBn-1Cn-Cn-2Cn-1-Cn-2,
n=2,3,4,,
1.12.14 b0 =A0
=C0,
b1 =B1,
a1 =A1-A0B1.

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=dndn-1an,
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/(bn-1bn)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+p11-p21+p2-p31+p3-pn1+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/cn-1)x1+(cn/cn-1)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+An-1wBn+Bn-1w,
Cn(0) =Cn,
Cn() =Cn-1
=An-1Bn-1.

§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 Cn-1,
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+b1b2-a2a3b4a3b4+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+b0b1b1-a1a2b3/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<12π-δ,
n=1,2,3,,

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

1.12.27 -12π+δ<phCn<12π-δ,
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.