1 Algebraic and Analytic Methods Areas 1.11 Zeros of Polynomials 1.13 Differential Equations

§1.12 Continued Fractions

Keywords:: continued fractions
Referenced by:: §3.10(i)
Permalink:: http://dlmf.nist.gov/1.12

§1.12(i) Notation
§1.12(ii) Convergents
§1.12(iii) Existence of Convergents
§1.12(iv) Contraction and Extension
§1.12(v) Convergence
§1.12(vi) Applications

§1.12(i) Notation

Keywords:: continued fractions
Permalink:: http://dlmf.nist.gov/1.12.i

The notation used throughout the DLMF for the continued fraction

1.12.1 $\cfracstyle{d}b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\raisebox{-6.0pt}{% $\ddots$}}}$

Permalink:: http://dlmf.nist.gov/1.12.E1
Encodings:: TeX, pMML, png

1.12.2 $b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+}}\cdots.$

Permalink:: http://dlmf.nist.gov/1.12.E2
Encodings:: TeX, pMML, png

§1.12(ii) Convergents

Notes:: See Jones and Thron (1980, pp. 20, 31–37). For (1.12.20)–(1.12.21), see Lorentzen and Waadeland (1992, pp. 8–9).
Keywords:: continued fractions
Referenced by:: §18.13
Permalink:: http://dlmf.nist.gov/1.12.ii

1.12.3 $C=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}},$ $a_{n}\not=0$ ,

Symbols:: : nonnegative integer and : continued fraction
A&S Ref:: 3.10.1
Referenced by:: §1.12(iii)
Permalink:: http://dlmf.nist.gov/1.12.E3
Encodings:: TeX, pMML, png

1.12.4 $C_{n}=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots\cfrac{a_{n}}{b_{n}}% }}=\frac{A_{n}}{B_{n}}.$

Symbols:: : nonnegative integer, $C_{n}$ : approximant, $A_{n}$ : th numerator and $B_{n}$ : th denominator
A&S Ref:: 3.10.1
Permalink:: http://dlmf.nist.gov/1.12.E4
Encodings:: TeX, pMML, png

$C_{n}$ is called the th approximant or convergent to . $A_{n}$ and $B_{n}$ are called the th (canonical) numerator and denominator respectively.

¶ Recurrence Relations

Keywords:: continued fractions

1.12.5

$A_{k}=b_{k}A_{{k-1}}+a_{k}A_{{k-2}}$ ,

$B_{k}=b_{k}B_{{k-1}}+a_{k}B_{{k-2}}$ , $k=1,2,3,\dots$ ,

Symbols:: : integer, $A_{n}$ : th numerator and $B_{n}$ : th denominator
A&S Ref:: 3.10.1
Referenced by:: ¶ ‣ §3.10(iii)
Permalink:: http://dlmf.nist.gov/1.12.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

1.12.6

$A_{{-1}}=1,$

$A_{0}=b_{0},$

$B_{{-1}}=0,$

$B_{0}=1.$

Symbols:: $A_{n}$ : th numerator and $B_{n}$ : th denominator
A&S Ref:: 3.10.1
Permalink:: http://dlmf.nist.gov/1.12.E6
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

¶ Determinant Formula

Keywords:: continued fractions

1.12.7 $A_{n}B_{{n-1}}-B_{n}A_{{n-1}}=(-1)^{{n-1}}\prod^{n}_{{k=1}}a_{k},$ $n=0,1,2,\dots$ .

Symbols:: : integer, : nonnegative integer, $A_{n}$ : th numerator and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E7
Encodings:: TeX, pMML, png

1.12.8 $C_{n}-C_{{n-1}}=\frac{(-1)^{{n-1}}\prod^{n}_{{k=1}}a_{k}}{B_{{n-1}}B_{n}},$ $n=1,2,3,\dots$ ,

Symbols:: : integer, : nonnegative integer, $C_{n}$ : approximant and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E8
Encodings:: TeX, pMML, png

1.12.9 $C_{n}=b_{0}+\frac{a_{1}}{B_{0}B_{1}}-\dots+(-1)^{{n-1}}\frac{\prod^{n}_{{k=1}}% a_{k}}{B_{{n-1}}B_{n}}.$

Symbols:: : integer, : nonnegative integer, $C_{n}$ : approximant and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E9
Encodings:: TeX, pMML, png

1.12.10 $a_{n}=\frac{A_{{n-1}}B_{n}-A_{n}B_{{n-1}}}{A_{{n-1}}B_{{n-2}}-A_{{n-2}}B_{{n-1% }}},$ $n=1,2,3,\dots$ ,

Symbols:: : nonnegative integer, $A_{n}$ : th numerator and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E10
Encodings:: TeX, pMML, png

1.12.11 $a_{n}=\frac{B_{n}}{B_{{n-2}}}\frac{C_{{n-1}}-C_{n}}{C_{{n-1}}-C_{{n-2}}},$ $n=2,3,4,\dots$ ,

Symbols:: : nonnegative integer, $C_{n}$ : approximant and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E11
Encodings:: TeX, pMML, png

1.12.12 $b_{n}=\frac{A_{n}B_{{n-2}}-A_{{n-2}}B_{n}}{A_{{n-1}}B_{{n-2}}-A_{{n-2}}B_{{n-1% }}},$ $n=1,2,3,\dots$ ,

Symbols:: : nonnegative integer, $A_{n}$ : th numerator and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E12
Encodings:: TeX, pMML, png

1.12.13 $b_{n}=\frac{B_{n}}{B_{{n-1}}}\frac{C_{n}-C_{{n-2}}}{C_{{n-1}}-C_{{n-2}}},$ $n=2,3,4,\dots$ ,

Symbols:: : nonnegative integer, $C_{n}$ : approximant and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E13
Encodings:: TeX, pMML, png

1.12.14

$b_{0}=A_{0}=C_{0},$

$b_{1}=B_{1},$

$a_{1}=A_{1}-A_{0}B_{1}.$

Symbols:: $C_{n}$ : approximant, $A_{n}$ : th numerator and $B_{n}$ : th denominator
Permalink:: http://dlmf.nist.gov/1.12.E14
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

¶ Equivalence

Keywords:: continued fractions

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

$b_{0}+\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}$ is equivalent to $b^{{\prime}}_{0}+\displaystyle{\cfrac{a^{{\prime}}_{1}}{b^{{\prime}}_{1}+% \cfrac{a^{{\prime}}_{2}}{b^{{\prime}}_{2}+\cdots}}}$ if there is a sequence $\{d_{n}\}^{\infty}_{{n=0}}$ , $d_{0}=1$ ,
$d_{n}\neq 0$ , such that

1.12.15 $a^{{\prime}}_{n}=d_{n}d_{{n-1}}a_{n},$ $n=1,2,3,\dots$ ,

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E15
Encodings:: TeX, pMML, png

and

1.12.16 $b^{{\prime}}_{n}=d_{n}b_{n},$ $n=0,1,2,\dots$ .

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E16
Encodings:: TeX, pMML, png

Formally,

1.12.17 $b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cdots}}}={b% _{0}+\cfrac{a_{1}/b_{1}}{1+\cfrac{a_{2}/(b_{1}b_{2})}{1+\cfrac{a_{3}/(b_{2}b_{% 3})}{1+\cdots\cfrac{a_{n}/(b_{{n-1}}b_{n})}{1+\cdots}}}}}={b_{0}+\cfrac{1}{(% \ifrac{1}{a_{1}})b_{1}+\cfrac{1}{(\ifrac{a_{1}}{a_{2}})b_{2}+\cfrac{1}{(\ifrac% {a_{2}}{(a_{1}a_{3})})b_{3}+\cfrac{1}{(\ifrac{a_{1}a_{3}}{(a_{2}a_{4})})b_{4}+% \cdots}}}}}.$

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E17
Encodings:: TeX, pMML, png

¶ Series

Keywords:: continued fractions

1.12.18 $p_{0}+\sum^{n}_{{k=1}}p_{1}p_{2}\cdots p_{k}=p_{0}+\cfrac{p_{1}}{1-\cfrac{p_{2% }}{1+p_{2}-\cfrac{p_{3}}{1+p_{3}-\cdots\cfrac{p_{n}}{1+p_{n}}}}},$ $n=0,1,2,\dots$ ,

Symbols:: : integer, : nonnegative integer and $p_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/1.12.E18
Encodings:: TeX, pMML, png

when $p_{k}\not=0$ , $k=1,2,3,\dots$ .

1.12.19 $\sum^{n}_{{k=0}}c_{k}x^{k}=c_{0}+\cfrac{c_{1}x}{1-\cfrac{(\ifrac{c_{2}}{c_{1}}% )x}{1+(\ifrac{c_{2}}{c_{1}})x-\cfrac{(\ifrac{c_{3}}{c_{2}})x}{1+(\ifrac{c_{3}}% {c_{2}})x-\cdots\cfrac{(\ifrac{c_{n}}{c_{{n-1}}})x}{1+(\ifrac{c_{n}}{c_{{n-1}}% })x}}}},$ $n=0,1,2,\dots$ ,

Symbols:: : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E19
Encodings:: TeX, pMML, png

when $c_{k}\not=0$ , $k=1,2,3,\dots$ .

¶ Fractional Transformations

Keywords:: continued fractions

Define

1.12.20 $C_{n}(w)=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots\frac{a_{n}}{b_{n% }+w}}}.$

Symbols:: : variable, : nonnegative integer and $C_{n}(w)$ : continued fraction
Referenced by:: §1.12(ii)
Permalink:: http://dlmf.nist.gov/1.12.E20
Encodings:: TeX, pMML, png

Then

1.12.21

$C_{n}(w)=\frac{A_{n}+A_{{n-1}}w}{B_{n}+B_{{n-1}}w},$

$C_{n}(0)=C_{n},$

$C_{n}(\infty)=C_{{n-1}}=\frac{A_{{n-1}}}{B_{{n-1}}}.$

Symbols:: : variable, : nonnegative integer, $A_{n}$ : th numerator, $B_{n}$ : th denominator and $C_{n}(w)$ : continued fraction
Referenced by:: §1.12(ii)
Permalink:: http://dlmf.nist.gov/1.12.E21
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

§1.12(iii) Existence of Convergents

Notes:: See Jones and Thron (1980, p. 34).
Keywords:: continued fractions
Permalink:: http://dlmf.nist.gov/1.12.iii

A sequence $\{C_{n}\}$ in the extended complex plane, $\Complex\cup\{\infty\}$ , can be a sequence of convergents of the continued fraction (1.12.3) iff

1.12.22

$C_{0}\not=\infty,$

$C_{n}\not=C_{{n-1}},$ $n=1,2,3,\dots$ .

Symbols:: : nonnegative integer and $C_{n}$ : approximant
Permalink:: http://dlmf.nist.gov/1.12.E22
Encodings:: TeX, TeX, pMML, pMML, png, png

§1.12(iv) Contraction and Extension

Notes:: See Jones and Thron (1980, pp. 42–43), or Lorentzen and Waadeland (1992, p. 84–85).
Keywords:: continued fractions
Referenced by:: §7.9
Permalink:: http://dlmf.nist.gov/1.12.iv

A contraction of a continued fraction is a continued fraction $C^{{\prime}}$ whose convergents $\{C^{{\prime}}_{n}\}$ form a subsequence of the convergents $\{C_{n}\}$ of . Conversely, is called an extension of $C^{{\prime}}$ . If $C^{{\prime}}_{n}=C_{{2n}}$ , $n=0,1,2,\dots$ , then $C^{{\prime}}$ is called the even part of . The even part of exists iff $b_{{2k}}\not=0$ , $k=1,2,\dots$ , and up to equivalence is given by

1.12.23 $b_{0}+\cfrac{a_{1}b_{2}}{a_{2}+b_{1}b_{2}-\cfrac{a_{2}a_{3}b_{4}}{a_{3}b_{4}+b% _{2}(a_{4}+b_{3}b_{4})-\cfrac{a_{4}a_{5}b_{2}b_{6}}{a_{5}b_{6}+b_{4}(a_{6}+b_{% 5}b_{6})-\cfrac{a_{6}a_{7}b_{4}b_{8}}{a_{7}b_{8}+b_{6}(a_{8}+b_{7}b_{8})-% \cdots}}}}.$

Permalink:: http://dlmf.nist.gov/1.12.E23
Encodings:: TeX, pMML, png

If $C^{{\prime}}_{n}=C_{{2n+1}}$ , $n=0,1,2,\dots$ , then $C^{{\prime}}$ is called the odd part of . The odd part of exists iff $b_{{2k+1}}\not=0$ , $k=0,1,2,\dots$ , and up to equivalence is given by

1.12.24 $\frac{a_{1}+b_{0}b_{1}}{b_{1}}-\cfrac{a_{1}a_{2}b_{3}/b_{1}}{a_{2}b_{3}+b_{1}(% a_{3}+b_{2}b_{3})-\cfrac{a_{3}a_{4}b_{1}b_{5}}{a_{4}b_{5}+b_{3}(a_{5}+b_{4}b_{% 5})-\cfrac{a_{5}a_{6}b_{3}b_{7}}{a_{6}b_{7}+b_{5}(a_{7}+b_{6}b_{7})-\cdots}}}.$

Permalink:: http://dlmf.nist.gov/1.12.E24
Encodings:: TeX, pMML, png

§1.12(v) Convergence

Notes:: See Jones and Thron (1980, pp. 88, 92) and Lorentzen and Waadeland (1992, pp. 30, 32).
Keywords:: continued fractions, convergence
Permalink:: http://dlmf.nist.gov/1.12.v

A continued fraction converges if the convergents $C_{n}$ tend to a finite limit as $n\to\infty$ .

¶ Pringsheim’s Theorem

Keywords:: Pringsheim’s theorem for continued fractions, continued fractions

The continued fraction $\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}$ converges when

1.12.25 $|b_{n}|\geq|a_{n}|+1,$ $n=1,2,3,\dots$ .

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E25
Encodings:: TeX, pMML, png

With these conditions the convergents $C_{n}$ satisfy $|C_{n}|<1$ and $C_{n}\to C$ with $|C|\leq 1$ .

¶ Van Vleck’s Theorem

Keywords:: Van Vleck’s theorem for continued fractions, continued fractions

Let the elements of the continued fraction $\displaystyle{\cfrac{1}{b_{1}+\cfrac{1}{b_{2}+\cdots}}}$ satisfy

1.12.26 $-\tfrac{1}{2}\pi+\delta<\mathop{\mathrm{ph}\/}\nolimits b_{n}<\tfrac{1}{2}\pi-\delta,$ $n=1,2,3,\dots$ ,

Symbols:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase and : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E26
Encodings:: TeX, pMML, png

where $\delta$ is an arbitrary small positive constant. Then the convergents $C_{n}$ satisfy

1.12.27 $-\tfrac{1}{2}\pi+\delta<\mathop{\mathrm{ph}\/}\nolimits C_{n}<\tfrac{1}{2}\pi-\delta,$ $n=1,2,3,\dots$ ,

Symbols:: $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : nonnegative integer and $C_{n}$ : approximant
Permalink:: http://dlmf.nist.gov/1.12.E27
Encodings:: TeX, pMML, png

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

1.12.28 $\sum^{\infty}_{{n=1}}|b_{n}|=\infty.$

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.12.E28
Encodings:: TeX, pMML, png

In this case $|\mathop{\mathrm{ph}\/}\nolimits C|\leq\tfrac{1}{2}\pi$ .

§1.12(vi) Applications

Keywords:: continued fractions
Permalink:: http://dlmf.nist.gov/1.12.vi

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

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