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

§1.12 Continued Fractions

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Keywords:: continued fractions
Referenced by:: §3.10(i)
Permalink:: http://dlmf.nist.gov/1.12
See also:: Annotations for Ch.1

§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

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Keywords:: continued fractions, notation
Permalink:: http://dlmf.nist.gov/1.12.i
See also:: Annotations for §1.12 and Ch.1

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 See also: Annotations for §1.12(i), §1.12 and Ch.1

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 See also: Annotations for §1.12(i), §1.12 and Ch.1

§1.12(ii) Convergents

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Keywords:: approximants, canonical denominator (or numerator), continued fractions, 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).
Referenced by:: §18.13
Permalink:: http://dlmf.nist.gov/1.12.ii
See also:: Annotations for §1.12 and Ch.1

1.12.3		$C=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}},$
$a_{n}\not=0$ ,
ⓘ Defines: $C$ : continued fraction (locally) Symbols: $n$ : nonnegative integer A&S Ref: 3.10.1 Referenced by: §1.12(iii) Permalink: http://dlmf.nist.gov/1.12.E3 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12 and Ch.1

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: $n$ : nonnegative integer, $A_{n}$ : $n$ th numerator, $B_{n}$ : $n$ th denominator and $C_{n}(w)$ : continued fraction A&S Ref: 3.10.1 Permalink: http://dlmf.nist.gov/1.12.E4 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12 and Ch.1

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

Recurrence Relations

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Keywords:: continued fractions, recurrence relations
See also:: Annotations for §1.12(ii), §1.12 and Ch.1

1.12.5		$\displaystyle A_{k}$	$\displaystyle=b_{k}A_{k-1}+a_{k}A_{k-2}$ ,
		$\displaystyle B_{k}$	$\displaystyle=b_{k}B_{k-1}+a_{k}B_{k-2}$ ,
	$k=1,2,3,\dots$ ,
ⓘ Symbols: $k$ : integer, $A_{n}$ : $n$ th numerator and $B_{n}$ : $n$ 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 See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

1.12.6	$\displaystyle A_{-1}$	$\displaystyle=1,$
	$\displaystyle A_{0}$	$\displaystyle=b_{0},$
	$\displaystyle B_{-1}$	$\displaystyle=0,$
	$\displaystyle B_{0}$	$\displaystyle=1.$
ⓘ Symbols: $A_{n}$ : $n$ th numerator and $B_{n}$ : $n$ 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 See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

Determinant Formula

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Keywords:: continued fractions, determinant formula
See also:: Annotations for §1.12(ii), §1.12 and Ch.1

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: $k$ : integer, $n$ : nonnegative integer, $A_{n}$ : $n$ th numerator and $B_{n}$ : $n$ th denominator Permalink: http://dlmf.nist.gov/1.12.E7 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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: $k$ : integer, $n$ : nonnegative integer, $B_{n}$ : $n$ th denominator and $C_{n}(w)$ : continued fraction Permalink: http://dlmf.nist.gov/1.12.E8 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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: $k$ : integer, $n$ : nonnegative integer, $B_{n}$ : $n$ th denominator and $C_{n}(w)$ : continued fraction Permalink: http://dlmf.nist.gov/1.12.E9 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

1.12.10	$\displaystyle a_{n}$	$\displaystyle=\frac{A_{n-1}B_{n}-A_{n}B_{n-1}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}},$
$n=1,2,3,\dots$ ,
ⓘ Symbols: $n$ : nonnegative integer, $A_{n}$ : $n$ th numerator and $B_{n}$ : $n$ th denominator Permalink: http://dlmf.nist.gov/1.12.E10 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1
1.12.11	$\displaystyle a_{n}$	$\displaystyle=\frac{B_{n}}{B_{n-2}}\frac{C_{n-1}-C_{n}}{C_{n-1}-C_{n-2}},$
$n=2,3,4,\dots$ ,
ⓘ Symbols: $n$ : nonnegative integer, $B_{n}$ : $n$ th denominator and $C_{n}(w)$ : continued fraction Permalink: http://dlmf.nist.gov/1.12.E11 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

1.12.12	$\displaystyle b_{n}$	$\displaystyle=\frac{A_{n}B_{n-2}-A_{n-2}B_{n}}{A_{n-1}B_{n-2}-A_{n-2}B_{n-1}},$
$n=1,2,3,\dots$ ,
ⓘ Symbols: $n$ : nonnegative integer, $A_{n}$ : $n$ th numerator and $B_{n}$ : $n$ th denominator Permalink: http://dlmf.nist.gov/1.12.E12 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1
1.12.13	$\displaystyle b_{n}$	$\displaystyle=\frac{B_{n}}{B_{n-1}}\frac{C_{n}-C_{n-2}}{C_{n-1}-C_{n-2}},$
$n=2,3,4,\dots$ ,
ⓘ Symbols: $n$ : nonnegative integer, $B_{n}$ : $n$ th denominator and $C_{n}(w)$ : continued fraction Permalink: http://dlmf.nist.gov/1.12.E13 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

1.12.14	$\displaystyle b_{0}$	$\displaystyle=A_{0}=C_{0},$
	$\displaystyle b_{1}$	$\displaystyle=B_{1},$
	$\displaystyle a_{1}$	$\displaystyle=A_{1}-A_{0}B_{1}.$
ⓘ Symbols: $A_{n}$ : $n$ th numerator, $B_{n}$ : $n$ th denominator and $C_{n}(w)$ : continued fraction Permalink: http://dlmf.nist.gov/1.12.E14 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

Equivalence

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Keywords:: continued fractions, equivalent
See also:: Annotations for §1.12(ii), §1.12 and Ch.1

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: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.12.E15 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

and

1.12.16		$b^{\prime}_{n}=d_{n}b_{n},$
$n=0,1,2,\dots$ .
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.12.E16 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.12.E17 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

Series

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Keywords:: continued fractions, series
See also:: Annotations for §1.12(ii), §1.12 and Ch.1

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: $k$ : integer, $n$ : nonnegative integer and $p_{k}$ : coefficients Permalink: http://dlmf.nist.gov/1.12.E18 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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: $k$ : integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.12.E19 Encodings: TeX, pMML, png See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

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

Fractional Transformations

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Keywords:: continued fractions, fractional transformations
See also:: Annotations for §1.12(ii), §1.12 and Ch.1

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: $w$ : variable, $n$ : 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 See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

Then

1.12.21	$\displaystyle C_{n}(w)$	$\displaystyle=\frac{A_{n}+A_{n-1}w}{B_{n}+B_{n-1}w},$
	$\displaystyle C_{n}(0)$	$\displaystyle=C_{n},$
	$\displaystyle C_{n}(\infty)$	$\displaystyle=C_{n-1}=\frac{A_{n-1}}{B_{n-1}}.$
ⓘ Symbols: $w$ : variable, $n$ : nonnegative integer, $A_{n}$ : $n$ th numerator, $B_{n}$ : $n$ 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 See also: Annotations for §1.12(ii), §1.12(ii), §1.12 and Ch.1

§1.12(iii) Existence of Convergents

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Keywords:: continued fractions, convergents, existence of
Notes:: See Jones and Thron (1980, p. 34).
Permalink:: http://dlmf.nist.gov/1.12.iii
See also:: Annotations for §1.12 and Ch.1

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

1.12.22		$\displaystyle C_{0}$	$\displaystyle\not=\infty,$
		$\displaystyle C_{n}$	$\displaystyle\not=C_{n-1},$
	$n=1,2,3,\dots$ .
ⓘ Symbols: $n$ : nonnegative integer and $C_{n}$ : approximant Permalink: http://dlmf.nist.gov/1.12.E22 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.12(iii), §1.12 and Ch.1

§1.12(iv) Contraction and Extension

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Keywords:: continued fractions, contraction, even part, extension, odd part
Notes:: See Jones and Thron (1980, pp. 42–43), or Lorentzen and Waadeland (1992, p. 84–85).
Referenced by:: §7.9
Permalink:: http://dlmf.nist.gov/1.12.iv
See also:: Annotations for §1.12 and Ch.1

A contraction of a continued fraction $C$ is a continued fraction $C^{\prime}$ whose convergents $\{C^{\prime}_{n}\}$ form a subsequence of the convergents $\{C_{n}\}$ of $C$ . Conversely, $C$ 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 $C$ . The even part of $C$ 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 See also: Annotations for §1.12(iv), §1.12 and Ch.1

If $C^{\prime}_{n}=C_{2n+1}$ , $n=0,1,2,\dots$ , then $C^{\prime}$ is called the odd part of $C$ . The odd part of $C$ 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 See also: Annotations for §1.12(iv), §1.12 and Ch.1

§1.12(v) Convergence

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Keywords:: continued fractions, convergence
Notes:: See Jones and Thron (1980, pp. 88, 92) and Lorentzen and Waadeland (1992, pp. 30, 32).
Permalink:: http://dlmf.nist.gov/1.12.v
See also:: Annotations for §1.12 and Ch.1

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

Pringsheim’s Theorem

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Keywords:: Pringsheim’s theorem, Pringsheim’s theorem for continued fractions, continued fractions
See also:: Annotations for §1.12(v), §1.12 and Ch.1

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: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.12.E25 Encodings: TeX, pMML, png See also: Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

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

Van Vleck’s Theorem

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Keywords:: Van Vleck’s theorem, Van Vleck’s theorem for continued fractions, continued fractions
See also:: Annotations for §1.12(v), §1.12 and Ch.1

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

1.12.26		$-\tfrac{1}{2}\pi+\delta<\operatorname{ph}b_{n}<\tfrac{1}{2}\pi-\delta,$
$n=1,2,3,\dots$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.12.E26 Encodings: TeX, pMML, png See also: Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

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

1.12.27		$-\tfrac{1}{2}\pi+\delta<\operatorname{ph}C_{n}<\tfrac{1}{2}\pi-\delta,$
$n=1,2,3,\dots$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $n$ : nonnegative integer and $C_{n}$ : approximant Permalink: http://dlmf.nist.gov/1.12.E27 Encodings: TeX, pMML, png See also: Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

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: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.12.E28 Encodings: TeX, pMML, png See also: Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

In this case $|\operatorname{ph}C|\leq\tfrac{1}{2}\pi$ .

§1.12(vi) Applications

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Keywords:: applications, continued fractions
Permalink:: http://dlmf.nist.gov/1.12.vi
See also:: Annotations for §1.12 and Ch.1

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