# Notations

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$!$
factorial (as in $n!$); Common Notations and Definitions
$!_{\NVar{q}}$
$q$-factorial (as in $n!_{q}$); (5.18.2)
$!!$
double factorial (as in $n!!$); Common Notations and Definitions
$\cdot$
$\mathbf{a}\cdot\mathbf{b}$: vector dot (or scalar) product; (1.6.2)
$*$
$f*g$: convolution for Fourier transforms; (1.14.5)
$\times$
$\mathbf{a}\times\mathbf{b}$: vector cross product; (1.6.9)
$\times$
$G\times H$: Cartesian product of groups $G$ and $H$; §23.1
$/$
$S_{1}/S_{2}$: set of all elements of $S_{1}$ modulo elements of $S_{2}$; §21.1
$\setminus$
set subtraction; Common Notations and Definitions
$\Longrightarrow$
implies; Common Notations and Definitions
$\Longleftrightarrow$
is equivalent to; Common Notations and Definitions
$\sim$
asymptotic equality; (2.1.1)
$\sim$
Poincaré asymptotic expansion; §2.1(iii)
$\nabla$
backward difference operator; §3.10(iii)
$\nabla$
del operator; (1.6.19)
$\nabla^{2}$
Laplacian for spherical coordinates; §1.5(ii)
$\nabla\NVar{f}$
gradient of differentiable scalar function $f$; (1.6.20)
$\nabla\cdot\NVar{\mathbf{F}}$
divergence of vector-valued function $\mathbf{F}$; (1.6.21)
$\nabla\times\NVar{\mathbf{F}}$
curl of vector-valued function $\mathbf{F}$; (1.6.22)
$\int$
integral; §1.4(iv)
$\int_{\NVar{a}}^{(\NVar{b}+)}$
loop integral in $\mathbb{C}$: path begins at $a$, encircles $b$ once in the positive sense, and returns to $a$.; §5.9(i)
$\int_{P}^{(1+,0+,1-,0-)}$
Pochhammer’s loop integral; §5.12
$\int\NVar{\cdots}{\mathrm{d}}_{\NVar{q}}\NVar{x}$
$q$-integral; §17.2(v)
$\pvint_{\NVar{a}}^{\NVar{b}}$
Cauchy principal value; (1.4.24)
$\NVar{f}(\NVar{c}-)$
limit on left (or from below); (1.4.3)
$\NVar{f}(\NVar{c}+)$
limit on right (or from above); (1.4.1)
$\overline{\NVar{z}}$
complex conjugate; (1.9.11)
${\NVar{x}}^{\underline{\NVar{n}}}$
falling factorial; §26.1
${\NVar{x}}^{\overline{\NVar{n}}}$
rising factorial; §26.1
$|\NVar{z}|$
modulus (or absolute value); (1.9.7)
$\|\NVar{\mathbf{a}}\|$
magnitude of vector; (1.6.3)
$\|\NVar{\mathbf{x}}\|_{2}$
Euclidean norm of a vector; §3.2(iii)
$\|\NVar{\mathbf{A}}\|_{p}$
$p$-norm of a matrix; §3.2(iii)
$\|\NVar{\mathbf{x}}\|_{p}$
$p$-norm of a vector; §3.2(iii)
$\|\NVar{\mathbf{x}}\|_{\infty}$
infinity (or maximum) norm of a vector; §3.2(iii)
$\scriptstyle\NVar{b_{0}}+\cfrac{\NVar{a_{1}}}{\NVar{b_{1}}+\cfrac{\NVar{a_{2}}% }{\NVar{b_{2}}+}}\cdots$
continued fraction; §1.12(i)
$\left\lceil\NVar{x}\right\rceil$
ceiling of $x$; Common Notations and Definitions
$\left\lfloor\NVar{x}\right\rfloor$
floor of $x$; Common Notations and Definitions
$[\NVar{z_{0},z_{1},\dots,z_{n}}]$
divided difference; §3.3(iii)
${\left[\NVar{a}\right]_{\NVar{\kappa}}}$
partitional shifted factorial; (35.4.1)
$\NVar{f}^{[\NVar{n}]}(\NVar{z})$
$n$th $q$-derivative; §17.2(iv)
$[\NVar{a},\NVar{b}]$
closed interval; Common Notations and Definitions
$[\NVar{a},\NVar{b})$
half-closed interval; Common Notations and Definitions
$[\NVar{a},\NVar{z}]!=\Gamma\left(a+1,z\right)$
notation used by Dingle (1973); §8.1
${[\NVar{p}/\NVar{q}]_{\NVar{f}}}$
$\left[\NVar{n}\atop\NVar{k}\right]=(-1)^{n-k}s\left(n,k\right)$
notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
$\left[\NVar{n}\atop\NVar{k}\right]$
Stirling cycle number; §26.13
$\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$
$q$-multinomial coefficient; §26.16
$\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$
$q$-binomial coefficient (or Gaussian polynomial); (17.2.27)
$(\NVar{z-1})!=\Gamma\left(z\right)$
alternative notation; §5.1
${\left(\NVar{a}\right)_{\NVar{n}}}$
Pochhammer’s symbol (or shifted factorial); §5.2(iii)
$(\NVar{a},\NVar{b})$
open interval; Common Notations and Definitions
$(\NVar{a},\NVar{b}]$
half-closed interval; Common Notations and Definitions
$(\NVar{a},\NVar{z})!=\gamma\left(a+1,z\right)$
notation used by Dingle (1973); §8.1
$\left(\NVar{m},\NVar{n}\right)$
greatest common divisor (gcd); §27.1
$(\NVar{n}|\NVar{P})$
Jacobi symbol; §27.9
$(\NVar{n}|\NVar{p})$
Legendre symbol; §27.9
$\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$
$q$-Pochhammer symbol (or $q$-shifted factorial); §17.2(i)
$\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$
multiple $q$-Pochhammer symbol; §17.2(i)
$\left(\NVar{j_{1}}\;\NVar{m_{1}}\;\NVar{j_{2}}\;\NVar{m_{2}}|\NVar{j_{1}}\;% \NVar{j_{2}}\;\NVar{j_{3}}\,\,\NVar{m_{3}}\right)$
Clebsch–Gordan coefficient; §34.1
$\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$
binomial coefficient; §1.2(i)
$\genfrac{(}{)}{0.0pt}{}{\NVar{n_{1}}+\NVar{n_{2}}+\dots+\NVar{n_{k}}}{\NVar{n_% {1}},\NVar{n_{2}},\ldots,\NVar{n_{k}}}$
multinomial coefficient; §26.4(i)
$\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$
$\mathit{3j}$ symbol; (34.2.4)
$\{\NVar{\ldots}\}$
sequence, asymptotic sequence (or scale), or enumerable set; §2.1(v)
$\left\{\NVar{z},\NVar{\zeta}\right\}$
Schwarzian derivative; (1.13.20)
$\left\{\NVar{n}\atop\NVar{k}\right\}=S\left(n,k\right)$
notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
$\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$
$\mathit{6j}$ symbol; (34.4.1)
$\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$
$\mathit{9j}$ symbol; (34.6.1)
$\left\langle\NVar{f},\NVar{\phi}\right\rangle$
tempered distribution; (2.6.11)
$\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$
inner-product of distribution; §1.16(i)
$\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$
Eulerian number; §26.14(i)