Notations

${\NVar{\mathbf{A}}}^{-1}$: matrix inverse; §1.2(vi)
$\mathbf{A}$: m by n matrix; §1.2(v)
$!$: factorial (as in $n!$ ); Common Notations and Definitions
$!_{\NVar{q}}$: $q$ -factorial (as in $n!_{q}$ ); (5.18.2)
$!!$: double factorial; Common Notations and Definitions
$\cdot$: $\mathbf{a}\cdot\mathbf{b}$ : vector dot (or scalar) product; (1.6.2)
$*$: $f*g$ : convolution product; (2.6.34)
$\times$: $\mathbf{a}\times\mathbf{b}$ : vector cross product; (1.6.9)
$\times$: $G\times H$ : Cartesian product of groups; §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
$\left|\NVar{z}\right|$: 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
$\left[\NVar{z_{0},z_{1},\dots,z_{n}}\right]\NVar{f}$: divided difference; (3.3.34)

${\left[\NVar{a}\right]_{\NVar{\kappa}}}$: 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
(with $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function)
${[\NVar{p}/\NVar{q}]_{\NVar{f}}}$: Padé approximant; §3.11(iv)
$\left[\NVar{n}\atop\NVar{k}\right]=(-1)^{n-k}s\left(n,k\right)$: notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
(with $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind)
$\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$: $q$ -multinomial coefficient; §26.16
$\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$: $q$ -binomial coefficient (or Gaussian polynomial); (17.2.27)
${\left(\NVar{S}\right)}$: cycle; §26.2
$(\NVar{z-1})!=\Gamma\left(z\right)$: alternative notation; §5.1
(with $\Gamma\left(\NVar{z}\right)$ : 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
$(\NVar{a},\NVar{z})!=\gamma\left(a+1,z\right)$: notation used by Dingle (1973); §8.1
(with $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function)
$\left(\NVar{m},\NVar{n}\right)$: 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
${\NVar{\mathbf{A}}}^{*}$: adjoint of matrix; §1.3(iv)
${\NVar{\mathbf{A}}}^{{\rm H}}$: Hermitian conjugate of matrix; (1.2.30)
$\NVar{\mathbf{A}}^{\mathrm{T}}$: transpose of matrix; (1.2.28)
$\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: 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)
$[{\NVar{\mathbf{A}}},{\NVar{\mathbf{B}}}]$: commutator; (1.2.66)
$\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{k}}$: Stirling cycle number of the first kind; (26.13.3)
$\left\langle\NVar{f},\NVar{g}\right\rangle$: inner product over functions; (1.18.12)
$\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$: inner product over vectors; (1.2.40)
$\left\|{\NVar{\mathbf{v}}}\right\|$: vector norm ( $l$ ²); (1.2.46)
$\{\NVar{\ldots}\}$: 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
(with $S\left(\NVar{n},\NVar{k}\right)$ : 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)
$\left\langle\NVar{f},\NVar{\phi}\right\rangle$: tempered distribution; (2.6.11)
$\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: action of distribution on test function; §1.16(i)
$\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$: Eulerian number; §26.14(i)

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