# §1.12 Continued Fractions

## §1.12(i) Notation

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

is

 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

 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

 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

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

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

 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

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

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

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

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

### Pringsheim’s Theorem

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

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

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