§1.2 Elementary Algebra

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

1.2.1		$$\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)=\frac{n!}{(n-k)!k!}=\left(\genfrac{}{}{0.0pt}{}{n}{n-k}\right).$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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), Erratum (V1.0.11) for Subsection 1.2(i) 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 $\left(\genfrac{}{}{0.0pt}{}{z}{k}\right)$ is defined via (1.2.6).

Let ${\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n$ be distinct constants, and $f(x)$ be a polynomial of degree less than $n$. Then

1.2.12		$$\frac{f(x)}{(x-{\alpha }_1)(x-{\alpha }_2)\mathrm{\cdots }(x-{\alpha }_n)}=\frac{A_1}{x-{\alpha }_1}+\frac{A_2}{x-{\alpha }_2}+\mathrm{\cdots }+\frac{A_n}{x-{\alpha }_n},$$
ⓘ 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		$$A_j=\frac{f({\alpha }_j)}{\prod _{k\ne j}({\alpha }_j-{\alpha }_k)}.$$
ⓘ 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		$$\frac{f(x)}{{(x-{\alpha }_1)}^n}=\frac{B_1}{x-{\alpha }_1}+\frac{B_2}{{(x-{\alpha }_1)}^2}+\mathrm{\cdots }+\frac{B_n}{{(x-{\alpha }_1)}^n},$$
ⓘ 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 $m_1,m_2,\mathrm{\dots },m_n$ are positive integers and , then there exist polynomials $f_j(x)$, , such that

1.2.16		$$\frac{f(x)}{{(x-{\alpha }_1)}^{m_1}{(x-{\alpha }_2)}^{m_2}\mathrm{\cdots }{(x-{\alpha }_n)}^{m_n}}=\frac{f_1(x)}{{(x-{\alpha }_1)}^{m_1}}+\frac{f_2(x)}{{(x-{\alpha }_2)}^{m_2}}+\mathrm{\cdots }+\frac{f_n(x)}{{(x-{\alpha }_n)}^{m_n}}.$$
ⓘ 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 $f_j(x)$, $j=1,2,\mathrm{\dots },n$, multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959a, pp. 151–159).

The arithmetic mean of $n$ numbers $a_1,a_2,\mathrm{\dots },a_n$ is

1.2.17		$$A=\frac{a_1+a_2+\mathrm{\cdots }+a_n}{n}.$$
ⓘ 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 $a_1,a_2,\mathrm{\dots },a_n$ are given by

1.2.18		$$G={(a_1{a}_2\mathrm{\cdots }{a}_n)}^{1/n},$$
ⓘ 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		$$\frac{1}{H}=\frac{1}{n}\left(\frac{1}{a_1}+\frac{1}{a_2}+\mathrm{\cdots }+\frac{1}{a_n}\right).$$
ⓘ 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 $a_1,a_2,\mathrm{\dots },a_n$, and $n$ positive numbers $p_1,p_2,\mathrm{\dots },p_n$ with

1.2.20		$$p_1+p_2+\mathrm{\cdots }+p_n=1,$$
ⓘ 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		$$M(r)={(p_1{a}_1^r+p_2{a}_2^r+\mathrm{\cdots }+p_n{a}_n^r)}^{1/r},$$
ⓘ 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		$$M(r)=0,$$
and $a_1{a}_2\mathrm{\dots }{a}_n=0$.
ⓘ 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		$\underset{r\to \mathrm{\infty }}{lim}M(r)$	$=\max (a_1,a_2,\mathrm{\dots },a_n),$
ⓘ 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		$\underset{r\to -\mathrm{\infty }}{lim}M(r)$	$=\min (a_1,a_2,\mathrm{\dots },a_n).$
ⓘ 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 $p_j=1/n$, $j=1,2,\mathrm{\dots },n$,

1.2.25		$M(1)$	$=A,$
		$M(-1)$	$=H,$
ⓘ 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		$$\underset{r\to 0}{lim}M(r)=G.$$
ⓘ 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 $j$.