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21: Mathematical Introduction

… ► ►►►►►

$ℂ$	complex plane (excluding infinity).
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empty products	unity.
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$n!$	factorial: $1\cdot 2\cdot 3\mathrm{\cdots }n$ if $n=1,2,3,\mathrm{\dots }$; 1 if $n=0$.
$n!!$	double factorial: $2\cdot 4\cdot 6\mathrm{\cdots }n$ if $n=2,4,6,\mathrm{\dots }$; $1\cdot 3\cdot 5\mathrm{\cdots }n$ if $n=1,3,5,\mathrm{\dots }$; 1 if $n=0,-1$.
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► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
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${\left(\alpha \right)}_n$	Pochhammer’s symbol: $\alpha (\alpha +1)(\alpha +2)\mathrm{\cdots }(\alpha +n-1)$ if $n=1,2,3,\mathrm{\dots }$; 1 if $n=0$.
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… ►Another numerical convention is that decimals followed by dots are unrounded; without the dots they are rounded. …

22: 35.4 Partitions and Zonal Polynomials

… ►Also, $|\kappa |$ denotes $k_1+\mathrm{\cdots }+k_m$, the weight of $\kappa$; $\mathrm{\ell }(\kappa )$ denotes the number of nonzero $k_j$; $a+\kappa$ denotes the vector $(a+k_1,\mathrm{\dots },a+k_m)$. … ►

35.4.1 $${\left[a\right]}_{\kappa }=\frac{{\mathrm{\Gamma }}_m\left(a+\kappa \right)}{{\mathrm{\Gamma }}_m\left(a\right)}=\prod _{j=1}^m{\left(a-\frac{1}{2}(j-1)\right)}_{k_j},$$

ⓘ

Defines:

${\left[a\right]}_{\kappa }$: partitional shifted factorial

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

►where ${\left(a\right)}_k=a(a+1)\mathrm{\cdots }(a+k-1)$. … ►

35.4.2

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Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer, $\kappa$: vector of nonnegative integers and $\mathrm{\ell }(\kappa )$: number of nonzeros

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

… ►

35.4.3 $$Z_{\kappa }\left(𝐓\right)=Z_{\kappa }\left(𝐈\right){\left|𝐓\right|}^{k_m}\underset{𝐎(m)}{\int }\prod _{j=1}^{m-1}{|{(𝐇𝐓{𝐇}^{-1})}_j|}^{k_j-k_{j+1}}\mathrm{d}𝐇,$$ $𝐓\in 𝓢$.

ⓘ

Symbols:

$\in$: element of, $\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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23: 18.2 General Orthogonal Polynomials

… ►

18.2.11_9

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Symbols:

$p_n(x)$: polynomial of degree $n$ and $x$: real variable

Referenced by:

§18.2(iv), §18.2(vi), Erratum (V1.2.0) Section 18.2

Permalink:

http://dlmf.nist.gov/18.2.E11_9

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

… ►

§18.2(v) Christoffel–Darboux Formula

… ►

18.2.18

ⓘ

Symbols:

$n$: nonnegative integer

Referenced by:

§18.2(vi)

Permalink:

http://dlmf.nist.gov/18.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(vi), §18.2 and Ch.18

… ►

18.2.27 $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$det$: determinant and $n$: nonnegative integer

Source:

Gautschi (2004, (2.1.1))

Referenced by:

§18.2(ix)

Permalink:

http://dlmf.nist.gov/18.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(ix), §18.2 and Ch.18

… ►

18.2.28 $n=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$det$: determinant and $n$: nonnegative integer

Source:

Gautschi (2004, (2.1.2))

Referenced by:

§18.2(ix)

Permalink:

http://dlmf.nist.gov/18.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(ix), §18.2 and Ch.18

…

24: 5.18 $q$-Gamma and $q$-Beta Functions

… ►

5.18.1 $${(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k),$$ $n=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $n$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(i), §5.18 and Ch.5

►

5.18.2 $$n{!}_q=1(1+q)\mathrm{\cdots }(1+q+\mathrm{\cdots }+q^{n-1})={(q;q)}_n{(1-q)}^{-n}.$$

ⓘ

Defines:

${!}_q$: $q$-factorial (as in $n{!}_q$)

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(i), §5.18 and Ch.5

… ►

5.18.3 $${(a;q)}_{\mathrm{\infty }}=\prod _{k=0}^{\mathrm{\infty }}(1-a{q}^k).$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(i), §5.18 and Ch.5

…

25: 1.11 Zeros of Polynomials

… ►

1.11.1 $$f(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\mathrm{\cdots }+a_0.$$

ⓘ

Symbols:

$z$: variable and $n$: nonnegative integer

Referenced by:

§1.11(i)

Permalink:

http://dlmf.nist.gov/1.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §1.11(i), §1.11(i), §1.11 and Ch.1

… ►Next, let $f(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\mathrm{\cdots }+a_0$. The zeros of $z^{n}f(1/z)=a_0{z}^n+a_1{z}^{n-1}+\mathrm{\cdots }+a_n$ are reciprocals of the zeros of $f(z)$. … ►

$⋮$

… ►The sum and product of the roots are respectively $-b/a$ and $c/a$. …

26: 23.17 Elementary Properties

… ►

23.17.4 $$\lambda \left(\tau \right)=16q(1-8q+44q^2+\mathrm{\cdots }),$$

ⓘ

Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $q$: nome and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

►

23.17.5 $$1728J\left(\tau \right)=q^{-2}+744+\mathrm{1\hspace{0.33em}96884}{q}^2+\mathrm{214\hspace{0.33em}93760}{q}^4+\mathrm{\cdots },$$

ⓘ

Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $q$: nome and $\tau$: complex variable

Referenced by:

§23.17(ii)

Permalink:

http://dlmf.nist.gov/23.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

… ►

§23.17(iii) Infinite Products

►

23.17.7 $$\lambda \left(\tau \right)=16q\prod _{n=1}^{\mathrm{\infty }}{\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)}^8,$$

ⓘ

Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $q$: nome, $n$: integer and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(iii), §23.17 and Ch.23

►

23.17.8 $$\eta \left(\tau \right)=q^{1/12}\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n}),$$

ⓘ

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.19

Permalink:

http://dlmf.nist.gov/23.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(iii), §23.17 and Ch.23

…

27: 3.8 Nonlinear Equations

… ►If $f(z_0)=f^{\prime }(z_0)=\mathrm{\cdots }=f^{(m-1)}(z_0)=0$ and $f^{(m)}(z_0)\ne 0$, then $z_0$ is a zero of $f$ of multiplicity $m$ ; compare §1.10(i). … ►After a zero $\zeta$ has been computed, the factor $z-\zeta$ is factored out of $p(z)$ as a by-product of Horner’s scheme (§1.11(i)) for the computation of $p(\zeta )$. … ►On the last iteration $q_n{z}^{n-2}+q_{n-1}{z}^{n-3}+\mathrm{\cdots }+q_2$ is the quotient on dividing $p(z)$ by $z^2-sz-t$. … ►

3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

ⓘ

Permalink:

http://dlmf.nist.gov/3.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

28: 29.12 Definitions

… ►

29.12.10

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer, $k$: real parameter and ${\xi }_n$: zeros

Permalink:

http://dlmf.nist.gov/29.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(iii), §29.12 and Ch.29

… ►

29.12.11

ⓘ

Defines:

$g(t_1,\mathrm{\dots },t_n)$: function (locally)

Symbols:

$n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, ($0$ or $\frac{1}{2}$): parameter $\rho$ , ($0$ or $\frac{1}{2}$): parameter $\sigma$ and ($0$ or $\frac{1}{2}$): parameter $\tau$

Referenced by:

§29.20(iii)

Permalink:

http://dlmf.nist.gov/29.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(iii), §29.12 and Ch.29

… ►

29.12.12 $$0\le t_1\le \mathrm{\cdots }\le t_m\le 1\le t_{m+1}\le \mathrm{\cdots }\le t_n\le k^{-2},$$

ⓘ

Symbols:

$m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.12(iii)

Permalink:

http://dlmf.nist.gov/29.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(iii), §29.12 and Ch.29

…

29: 19.16 Definitions

… ►

19.16.9 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a^{\prime }-1}\prod _{j=1}^n{(t+z_j)}^{-b_j}dt=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a-1}\prod _{j=1}^n{(1+t{z}_j)}^{-b_j}dt,$$ $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$, $z_j\in ℂ\setminus (-\mathrm{\infty },0]$,

ⓘ

Defines:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function

Symbols:

$\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $ℝ$: real line, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9)

Permalink:

http://dlmf.nist.gov/19.16.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.18):

The constraint $a,a^{\prime }>0$, was replaced with $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$. It therefore follows from (19.16.10) that $a^{\prime }>0$.

Suggested 2017-07-25 by Bastien Roucariès

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

…

30: 26.15 Permutations: Matrix Notation

… ►The inversion number of $\sigma$ is a sum of products of pairs of entries in the matrix representation of $\sigma$: … ►The Ferrers board of shape $(b_1,b_2,\mathrm{\dots },b_n)$, $0\le b_1\le b_2\le \mathrm{\cdots }\le b_n$, is the set $B=\{(j,k)|\mathrm{\hspace{0.25em}1}\le j\le n,1\le k\le b_j\}$. … ►

26.15.11 $$\sum _{k=0}^n{r}_{n-k}(B){\left(x-k+1\right)}_k=\prod _{j=1}^n(x+b_j-j+1).$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.15, §26.15 and Ch.26

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