§1.2 Elementary Algebra

§1.2(i) Binomial Coefficients

In (1.2.1) and (1.2.3) $k$ and $n$ are nonnegative integers and $k\leq n$. In (1.2.2), (1.2.4), and (1.2.5) $n$ is a positive integer. See also §26.3(i).

 1.2.1 $\genfrac{(}{)}{0.0pt}{}{n}{k}=\frac{n!}{(n-k)!k!}=\genfrac{(}{)}{0.0pt}{}{n}{n% -k}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $!$: factorial (as in $n!$), $k$: integer and $n$: nonnegative integer A&S Ref: 3.1.2 Referenced by: §1.2(i), §1.2(i), 1st Erratum Permalink: http://dlmf.nist.gov/1.2.E1 Encodings: TeX, pMML, png See also: Annotations for §1.2(i), §1.2 and Ch.1

For complex $z$ the binomial coefficient $\genfrac{(}{)}{0.0pt}{}{z}{k}$ is defined via (1.2.6).

Binomial Theorem

 1.2.2 $(a+b)^{n}=a^{n}+\genfrac{(}{)}{0.0pt}{}{n}{1}a^{n-1}b+\genfrac{(}{)}{0.0pt}{}{% n}{2}a^{n-2}b^{2}+\dots+\genfrac{(}{)}{0.0pt}{}{n}{n-1}ab^{n-1}+b^{n}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient and $n$: nonnegative integer A&S Ref: 3.1.1 Referenced by: §1.10(i), §1.10(i), §1.2(i), §1.2(i), §4.6(ii), §4.6(ii), Erratum Subsection 1.2(i) Permalink: http://dlmf.nist.gov/1.2.E2 Encodings: TeX, pMML, png See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1
 1.2.3 $\genfrac{(}{)}{0.0pt}{}{n}{0}+\genfrac{(}{)}{0.0pt}{}{n}{1}+\dots+\genfrac{(}{% )}{0.0pt}{}{n}{n}=2^{n}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient and $n$: nonnegative integer A&S Ref: 3.1.6 Referenced by: §1.2(i) Permalink: http://dlmf.nist.gov/1.2.E3 Encodings: TeX, pMML, png See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1
 1.2.4 $\genfrac{(}{)}{0.0pt}{}{n}{0}-\genfrac{(}{)}{0.0pt}{}{n}{1}+\dots+(-1)^{n}% \genfrac{(}{)}{0.0pt}{}{n}{n}=0.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient and $n$: nonnegative integer A&S Ref: 3.1.7 Referenced by: §1.2(i), §1.2(i), Erratum Subsection 1.2(i) Permalink: http://dlmf.nist.gov/1.2.E4 Encodings: TeX, pMML, png See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1
 1.2.5 $\genfrac{(}{)}{0.0pt}{}{n}{0}+\genfrac{(}{)}{0.0pt}{}{n}{2}+\genfrac{(}{)}{0.0% pt}{}{n}{4}+\dots+\genfrac{(}{)}{0.0pt}{}{n}{\ell}=2^{n-1},$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\ell$: integer and $n$: nonnegative integer Referenced by: §1.2(i), §1.2(i), Erratum Subsection 1.2(i) Permalink: http://dlmf.nist.gov/1.2.E5 Encodings: TeX, pMML, png See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

where $\ell$ is $n$ or $n-1$ according as $n$ is even or odd.

In (1.2.6)–(1.2.9) $k$ and $m$ are nonnegative integers and $z$ is complex.

 1.2.6 $\genfrac{(}{)}{0.0pt}{}{z}{k}=\frac{z(z-1)\cdots(z-k+1)}{k!}=\frac{(-1)^{k}{% \left(-z\right)_{k}}}{k!}=(-1)^{k}\genfrac{(}{)}{0.0pt}{}{k-z-1}{k}.$ ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $!$: factorial (as in $n!$), $z$: variable, $k$: integer and $n$: nonnegative integer A&S Ref: 3.1.3 Referenced by: §1.10(i), §1.10(i), §1.2(i), §1.2(i), §1.2(i), (25.11.7), §4.6(ii), §4.6(ii), 1st Erratum Permalink: http://dlmf.nist.gov/1.2.E6 Encodings: TeX, pMML, png Clarification (effective with 1.0.11): As a notational clarification, wherever $n$ appeared originally in in this equation, it has been replaced by $z$. See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1
 1.2.7 $\genfrac{(}{)}{0.0pt}{}{z+1}{k}=\genfrac{(}{)}{0.0pt}{}{z}{k}+\genfrac{(}{)}{0% .0pt}{}{z}{k-1}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $z$: variable, $k$: integer and $n$: nonnegative integer A&S Ref: 3.1.4 Referenced by: §1.2(i) Permalink: http://dlmf.nist.gov/1.2.E7 Encodings: TeX, pMML, png Clarification (effective with 1.0.11): As a notational clarification, wherever $n$ appeared originally in in this equation, it has been replaced by $z$. See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1
 1.2.8 $\sum^{m}_{k=0}\genfrac{(}{)}{0.0pt}{}{z+k}{k}=\genfrac{(}{)}{0.0pt}{}{z+m+1}{m}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $z$: variable, $k$: integer, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.2.E8 Encodings: TeX, pMML, png Clarification (effective with 1.0.11): As a notational clarification, wherever $n$ appeared originally in in this equation, it has been replaced by $z$. See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1
 1.2.9 $\genfrac{(}{)}{0.0pt}{}{z}{0}-\genfrac{(}{)}{0.0pt}{}{z}{1}+\dots+(-1)^{m}% \genfrac{(}{)}{0.0pt}{}{z}{m}=(-1)^{m}\genfrac{(}{)}{0.0pt}{}{z-1}{m}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $z$: variable, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §1.2(i), §1.2(i), 1st Erratum Permalink: http://dlmf.nist.gov/1.2.E9 Encodings: TeX, pMML, png Clarification (effective with 1.0.11): As a notational clarification, wherever $n$ appeared originally in in this equation, it has been replaced by $z$. See also: Annotations for §1.2(i), §1.2(i), §1.2 and Ch.1

§1.2(ii) Finite Series

Arithmetic Progression

 1.2.10 $a+(a+d)+(a+2d)+\dots+(a+(n-1)d)=na+\tfrac{1}{2}n(n-1)d=\tfrac{1}{2}n(a+\ell),$ ⓘ Symbols: $n$: nonnegative integer A&S Ref: 3.1.9 Permalink: http://dlmf.nist.gov/1.2.E10 Encodings: TeX, pMML, png See also: Annotations for §1.2(ii), §1.2(ii), §1.2 and Ch.1

where $\ell$ = last term of the series = $a+(n-1)d$.

Geometric Progression

 1.2.11 $a+ax+ax^{2}+\dots+ax^{n-1}=\frac{a(1-x^{n})}{1-x},$ $x\not=1$. ⓘ Symbols: $n$: nonnegative integer A&S Ref: 3.1.10 Permalink: http://dlmf.nist.gov/1.2.E11 Encodings: TeX, pMML, png See also: Annotations for §1.2(ii), §1.2(ii), §1.2 and Ch.1

§1.2(iii) Partial Fractions

Let $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ be distinct constants, and $f(x)$ be a polynomial of degree less than $n$. Then

 1.2.12 $\frac{f(x)}{(x-\alpha_{1})(x-\alpha_{2})\cdots(x-\alpha_{n})}=\frac{A_{1}}{x-% \alpha_{1}}+\frac{A_{2}}{x-\alpha_{2}}+\dots+\frac{A_{n}}{x-\alpha_{n}},$ ⓘ Symbols: $n$: nonnegative integer, $f(x)$: polynomial of degree less than $n$ and $A_{j}$: coefficient Permalink: http://dlmf.nist.gov/1.2.E12 Encodings: TeX, pMML, png See also: Annotations for §1.2(iii), §1.2 and Ch.1

where

 1.2.13 $A_{j}=\frac{f(\alpha_{j})}{\prod\limits_{k\not=j}(\alpha_{j}-\alpha_{k})}.$ ⓘ Defines: $A_{j}$: coefficient (locally) Symbols: $j$: integer, $k$: integer and $f(x)$: polynomial of degree less than $n$ Permalink: http://dlmf.nist.gov/1.2.E13 Encodings: TeX, pMML, png See also: Annotations for §1.2(iii), §1.2 and Ch.1

Also,

 1.2.14 $\frac{f(x)}{(x-\alpha_{1})^{n}}=\frac{B_{1}}{x-\alpha_{1}}+\frac{B_{2}}{(x-% \alpha_{1})^{2}}+\dots+\frac{B_{n}}{(x-\alpha_{1})^{n}},$ ⓘ Symbols: $n$: nonnegative integer, $f(x)$: polynomial of degree less than $n$ and $B_{j}$: coefficient Permalink: http://dlmf.nist.gov/1.2.E14 Encodings: TeX, pMML, png See also: Annotations for §1.2(iii), §1.2 and Ch.1

where

 1.2.15 $B_{j}=\frac{f^{(n-j)}(\alpha_{1})}{(n-j)!},$ ⓘ Defines: $B_{j}$: coefficient (locally) Symbols: $!$: factorial (as in $n!$), $j$: integer, $n$: nonnegative integer and $f(x)$: polynomial of degree less than $n$ Permalink: http://dlmf.nist.gov/1.2.E15 Encodings: TeX, pMML, png See also: Annotations for §1.2(iii), §1.2 and Ch.1

and $f^{(k)}$ is the $k$-th derivative of $f$1.4(iii)).

If $m_{1},m_{2},\dots,m_{n}$ are positive integers and $\deg f<\sum_{j=1}^{n}m_{j}$, then there exist polynomials $f_{j}(x)$, $\deg f_{j}, such that

 1.2.16 $\frac{f(x)}{(x-\alpha_{1})^{m_{1}}(x-\alpha_{2})^{m_{2}}\cdots(x-\alpha_{n})^{% m_{n}}}=\frac{f_{1}(x)}{(x-\alpha_{1})^{m_{1}}}+\frac{f_{2}(x)}{(x-\alpha_{2})% ^{m_{2}}}+\cdots+\frac{f_{n}(x)}{(x-\alpha_{n})^{m_{n}}}.$ ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer and $f(x)$: polynomial of degree less than $n$ Permalink: http://dlmf.nist.gov/1.2.E16 Encodings: TeX, pMML, png See also: Annotations for §1.2(iii), §1.2 and Ch.1

To find the polynomials $f_{j}(x)$, $j=1,2,\dots,n$, multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959a, pp. 151–159).

§1.2(iv) Means

The arithmetic mean of $n$ numbers $a_{1},a_{2},\dots,a_{n}$ is

 1.2.17 $A=\frac{a_{1}+a_{2}+\dots+a_{n}}{n}.$ ⓘ Defines: $A$: arithmetic mean (locally) Symbols: $n$: nonnegative integer A&S Ref: 3.1.11 Permalink: http://dlmf.nist.gov/1.2.E17 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1

The geometric mean $G$ and harmonic mean $H$ of $n$ positive numbers $a_{1},a_{2},\dots,a_{n}$ are given by

 1.2.18 $G=(a_{1}a_{2}\cdots a_{n})^{1/n},$ ⓘ Defines: $G$: geometric mean (locally) Symbols: $n$: nonnegative integer A&S Ref: 3.1.12 Permalink: http://dlmf.nist.gov/1.2.E18 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1
 1.2.19 $\frac{1}{H}=\frac{1}{n}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\dots+\frac{1}{a_% {n}}\right).$ ⓘ Defines: $H$: harmonic mean (locally) Symbols: $n$: nonnegative integer A&S Ref: 3.1.13 Permalink: http://dlmf.nist.gov/1.2.E19 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1

If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_{1},a_{2},\dots,a_{n}$, and $n$ positive numbers $p_{1},p_{2},\dots,p_{n}$ with

 1.2.20 $p_{1}+p_{2}+\dots+p_{n}=1,$ ⓘ Symbols: $n$: nonnegative integer and $p_{j}$; positive numbers Permalink: http://dlmf.nist.gov/1.2.E20 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1

is defined by

 1.2.21 $M(r)=(p_{1}a_{1}^{r}+p_{2}a_{2}^{r}+\dots+p_{n}a_{n}^{r})^{1/r},$ ⓘ Defines: $M(r)$: weighted mean (locally) Symbols: $n$: nonnegative integer and $p_{j}$; positive numbers Permalink: http://dlmf.nist.gov/1.2.E21 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1

with the exception

 1.2.22 $M(r)=0,$ $r<0$ and $a_{1}a_{2}\dots a_{n}=0$. ⓘ Symbols: $n$: nonnegative integer and $M(r)$: weighted mean A&S Ref: 3.1.15 Permalink: http://dlmf.nist.gov/1.2.E22 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1
 1.2.23 $\displaystyle\lim_{r\to\infty}M(r)$ $\displaystyle=\max(a_{1},a_{2},\dots,a_{n}),$ ⓘ Symbols: $n$: nonnegative integer and $M(r)$: weighted mean A&S Ref: 3.1.16 Permalink: http://dlmf.nist.gov/1.2.E23 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1 1.2.24 $\displaystyle\lim_{r\to-\infty}M(r)$ $\displaystyle=\min(a_{1},a_{2},\dots,a_{n}).$ ⓘ Symbols: $n$: nonnegative integer and $M(r)$: weighted mean A&S Ref: 3.1.17 Permalink: http://dlmf.nist.gov/1.2.E24 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1

For $p_{j}=1/n$, $j=1,2,\dots,n$,

 1.2.25 $\displaystyle M(1)$ $\displaystyle=A,$ $\displaystyle M(-1)$ $\displaystyle=H,$ ⓘ Symbols: $A$: arithmetic mean, $H$: harmonic mean and $M(r)$: weighted mean A&S Ref: 3.1.19 3.1.20 Permalink: http://dlmf.nist.gov/1.2.E25 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.2(iv), §1.2 and Ch.1

and

 1.2.26 $\lim_{r\to 0}M(r)=G.$ ⓘ Symbols: $G$: geometric mean and $M(r)$: weighted mean A&S Ref: 3.1.18 Permalink: http://dlmf.nist.gov/1.2.E26 Encodings: TeX, pMML, png See also: Annotations for §1.2(iv), §1.2 and Ch.1

The last two equations require $a_{j}>0$ for all $j$.