# Notations

♦*♦ABCDEFGHIJKLMNOPQRSTUVWXYZ
$!$
factorial (as in $n!$); Common Notations and Definitions
${}{!}_{q}$
$q$-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 Laplace transforms; 1.14.30
$*$
$f*g$: convolution for Mellin transforms; 1.14.39
$*$
$f*g$: convolution product; 2.6.34
$*$
$f*g$: convolution for Fourier transforms; 1.14.5
$\times$
$G\times H$: Cartesian product of groups $G$ and $H$; §23.1
$\times$
$\mathbf{a}\times\mathbf{b}$: vector cross product; 1.6.9
$\Longrightarrow$
implies; Common Notations and Definitions
$\Longleftrightarrow$
is equivalent to; Common Notations and Definitions
$\setmod$
$S_{1}\setmod S_{2}$: set of all elements of $S_{1}$ modulo elements of $S_{2}$; §21.1
$\setminus$
set subtraction; Common Notations and Definitions
$\sim$
asymptotic equality; 2.1.1
$\nabla$
backward difference operator; §3.10(iii)
$\nabla$
del operator; 1.6.19
$\nabla^{2}$
Laplacian; §1.5(ii)
$\nabla^{2}$
Laplacian for polar coordinates; §1.5(ii)
$\nabla^{2}$
Laplacian for cylindrical coordinates; §1.5(ii)
$\nabla^{2}$
Laplacian for spherical coordinates; §1.5(ii)
$\nabla f$
gradient of differentiable scalar function $f$; 1.6.20
$\nabla\cdot\mathbf{F}$
divergence of vector-valued function $\mathbf{F}$; 1.6.21
$\nabla\times\mathbf{F}$
curl of vector-valued function $\mathbf{F}$; 1.6.22
$\int$
integral; §1.4(iv)
$\pvint_{a}^{b}$
Cauchy principal value; 1.4.24
$\int_{a}^{(b+)}$
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
$\int\cdots{d}_{q}x$
$q$-integral; §17.2(v)
$\conj{z}$
complex conjugate; 1.9.11
$|z|$
modulus (or absolute value); 1.9.7
$\|\mathbf{a}\|$
magnitude of vector; 1.6.3
$\|\mathbf{A}\|_{p}$
$p$-norm of a matrix; §3.2(iii)
$\|\mathbf{x}\|_{2}$
Euclidean norm of a vector; §3.2(iii)
$\|\mathbf{x}\|_{\infty}$
infinity (or maximum) norm of a vector; §3.2(iii)
$\|\mathbf{x}\|_{p}$
$p$-norm of a vector; §3.2(iii)
$f(c+)$
limit on right (or from above); 1.4.1
$f(c-)$
limit on left (or from below); 1.4.3
$f^{[n]}(z)$
$n$th $q$-derivative; §17.2(iv)
${x}^{\underline{n}}$
falling factorial; §26.1
${x}^{\overline{n}}$
rising factorial; §26.1
$\scriptstyle b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+}}\cdots$
continued fraction; §1.12(i)
$\left(a\right)_{n}$
Pochhammer’s symbol; §5.2(iii)
$(z-1)!=\mathop{\Gamma\/}\nolimits\!\left(z\right)$
alternative notation; §5.1
$\left(m,n\right)$
greatest common divisor (gcd); §27.1
$(a,b)$
open interval; Common Notations and Definitions
$(a,b]$
half-closed interval; Common Notations and Definitions
$\mathop{(n|P)\/}\nolimits$
Jacobi symbol; §27.9
$\mathop{(n|p)\/}\nolimits$
Legendre symbol; §27.9
$\left(a;q\right)_{n}$
$q$-Pochhammer (or $q$-shifted factorial); §17.2(i)
$\left(a_{1},a_{2},\dots,a_{r};q\right)_{n}$
multiple $q$-Pochhammer symbol; §17.2(i)
$(a,z)!=\mathop{\gamma\/}\nolimits\!\left(a+1,z\right)$
notation used by Dingle (1973); §8.1
$\left(j_{1}m_{1}j_{2}m_{2}|j_{1}j_{2}j_{3}-m_{3}\right)$
Clebsch–Gordan coefficient; §34.1
${\left({{m}\atop{n}}\right)}$
binomial coefficient; 1.2.1
${\left({{n_{1}+n_{2}+\dots+n_{k}}\atop{n_{1},n_{2},\ldots,n_{k}}}\right)}$
multinomial coefficient; §26.4(i)
$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$
$\mathit{3j}$ symbol; 34.2.4
$\left\langle\Lambda,\phi\right\rangle$
distribution; §1.16(i)
$\left\langle f,\phi\right\rangle$
tempered distribution; 2.6.11
$\left\langle\delta,\phi\right\rangle$
Dirac delta distribution; §1.16(iii)
$\mathop{\genfrac{<}{>}{0.0pt}{}{n}{k}\/}\nolimits$
Eulerian number; §26.14(i)
$\left\lfloor x\right\rfloor$
floor of $x$; Common Notations and Definitions
$\left\lceil x\right\rceil$
ceiling of $x$; Common Notations and Definitions
$[z_{0},z_{1},\dots,z_{n}]$
divided difference; §3.3(iii)
$\left[a\right]_{\kappa}$
partitional shifted factorial; 35.4.1
$[a,b)$
half-closed interval; Common Notations and Definitions
$[a,b]$
closed interval; Common Notations and Definitions
$\mathop{{[p/q]_{f}}\/}\nolimits$
$[a,z]!=\mathop{\Gamma\/}\nolimits\!\left(a+1,z\right)$
notation used by Dingle (1973); §8.1
$\left[n\atop k\right]$
Stirling cycle number; §26.13
$\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}$
$q$-binomial coefficient (or Gaussian polynomial); 17.2.27
$\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\dots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q}$
$q$-multinomial coefficient; §26.16
$\left[n\atop k\right]=(-1)^{n-k}\mathop{s\/}\nolimits\!\left(n,k\right)$
notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
$\{\ldots\}$
sequence, asymptotic sequence (or scale), or enumerable set; §2.1(v)
$\left\{z,\zeta\right\}$
Schwarzian derivative; 1.13.20
$\left\{n\atop k\right\}=\mathop{S\/}\nolimits\!\left(n,k\right)$
notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}$
$\mathit{6j}$ symbol; 34.4.1
$\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}$
$\mathit{9j}$ symbol; 34.6.1