Notations

♦*♦A♦B♦C♦D♦E♦F♦G♦H♦I♦J♦K♦L♦M♦N♦O♦P♦Q♦R♦S♦T♦U♦V♦W♦X♦Y♦Z♦

$!$: 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
$\setmod$: $S_{1}\setmod S_{2}$ : 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 $\Complex$ : 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
$\int\NVar{\cdots}{d}_{\NVar{q}}\NVar{x}$: $q$ -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
$\conj{\NVar{z}}$: 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
$[\NVar{a},\NVar{z}]!=\mathop{\Gamma\/}\nolimits\!\left(a+1,z\right)$: notation used by Dingle (1973); §8.1
(with $\mathop{\Gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function)
$\mathop{{[\NVar{p}/\NVar{q}]_{\NVar{f}}}\/}\nolimits$: Padé approximant; §3.11(iv)
$\left[\NVar{n}\atop\NVar{k}\right]=(-1)^{n-k}\mathop{s\/}\nolimits\!\left(n,k\right)$: notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
(with $\mathop{s\/}\nolimits\!\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind)
$\left[\NVar{n}\atop\NVar{k}\right]$: 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
$(\NVar{z-1})!=\mathop{\Gamma\/}\nolimits\!\left(z\right)$: alternative notation; §5.1
(with $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$ : gamma function)
$\left(\NVar{a}\right)_{\NVar{n}}$: 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
$(\NVar{a},\NVar{z})!=\mathop{\gamma\/}\nolimits\!\left(a+1,z\right)$: notation used by Dingle (1973); §8.1
(with $\mathop{\gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function)
$\left(\NVar{m},\NVar{n}\right)$: greatest common divisor (gcd); §27.1
$\mathop{(\NVar{n}|\NVar{P})\/}\nolimits$: Jacobi symbol; §27.9
$\mathop{(\NVar{n}|\NVar{p})\/}\nolimits$: 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
$\binom{\NVar{m}}{\NVar{n}}$: binomial coefficient; §1.2(i)
${\left({{\NVar{n_{1}}+\NVar{n_{2}}+\dots+\NVar{n_{k}}}\atop{\NVar{n_{1}},\NVar% {n_{2}},\ldots,\NVar{n_{k}}}}\right)}$: 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
$\left\{\NVar{n}\atop\NVar{k}\right\}=\mathop{S\/}\nolimits\!\left(n,k\right)$: notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
(with $\mathop{S\/}\nolimits\!\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind)
$\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$: $\mathit{6j}$ 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
$\left\langle\NVar{\delta},\NVar{\phi}\right\rangle$: Dirac delta distribution; §1.16(iii)
$\left\langle\NVar{f},\NVar{\phi}\right\rangle$: tempered distribution; 2.6.11
$\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: distribution; §1.16(i)
$\mathop{\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}\/}\nolimits$: Eulerian number; §26.14(i)