DLMF: Notations *

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♦

$!$: $n!$ : factorial; Common Notations and Definitions
$!_{q}$: $n!_{q}$ : $q$ -factorial; §5.18(i)
$!!$: $n!!$ : double factorial; 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)
$*$: $f*g$ : convolution for Laplace transforms; (1.14.30)
$*$: $f*g$ : convolution product; (2.6.34)
$*$: $f*g$ : convolution for Mellin transforms; (1.14.39)
$\times$: $G\times H$ : 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 for polar coordinates; ¶ ‣ §1.5(ii)
$\nabla^{2}$: Laplacian; ¶ ‣ §1.5(ii)
$\nabla^{2}$: Laplacian for spherical coordinates; ¶ ‣ §1.5(ii)
$\nabla^{2}$: Laplacian for cylindrical 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
(with $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function)
$(a,b)$: 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
$\left(a;q\right)_{{n}}$: $q$ -factorial (or $q$ -shifted factorial); §17.2(i)
$\left(a;q\right)_{{n}}$: $q$ -factorial (or $q$ -shifted factorial); §5.18(i)
$\left(a;q\right)_{{\nu}}$: $q$ -shifted factorial (generalized); §17.2(i)
$\left(a;q\right)_{{\infty}}$: $q$ -shifted factorial; §17.2(i)
$\left(a_{1},a_{2},\dots,a_{r};q\right)_{{n}}$: multiple $q$ -shifted factorial; §17.2(i)
$\left(a_{1},a_{2},\dots,a_{r};q\right)_{{\infty}}$: multiple $q$ -shifted factorial; §17.2(i)
$(a,z)!=\mathop{\gamma\/}\nolimits\!\left(a+1,z\right)$: notation used by Dingle (1973); §8.1
(with $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function)
$\left(j_{1}m_{1}j_{2}m_{2}|j_{1}j_{2}j_{3}-m_{3}\right)$: Clebsch–Gordan coefficient; §34.1
$\binom{m}{n}$: binomial coefficient; §26.3(i)
$\binom{m}{n}$: binomial coefficient; (1.2.1)
$\multinomial{n_{1}+n_{2}+\dots+n_{k}}{n_{1},n_{2},\ldots,n_{k}}$: multinomial coefficient; §26.4(i)
$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$: $3j$ symbol; (34.2.4)
$\left\langle f,\phi\right\rangle$: tempered distribution; (2.6.11)
$\left\langle\Lambda,\phi\right\rangle$: distribution; §1.16(i)
$\left\langle\delta,\phi\right\rangle$: 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
$\mathop{{[p/q]_{{f}}}\/}\nolimits$: Padé approximant; §3.11(iv)
$[a,z]!=\mathop{\Gamma\/}\nolimits\!\left(a+1,z\right)$: notation used by Dingle (1973); §8.1
(with $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function)
$\left[n\atop k\right]$: Stirling cycle number; §26.13
$\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$: $q$ -binomial coefficient (or Gaussian polynomial); (17.2.27)
$\genfrac{[}{]}{0.0pt}{}{n}{m}_{{q}}$: $q$ -binomial coefficient (or Gaussian polynomial); §26.9(ii)
$\left[n\atop k\right]=\mathop{s\/}\nolimits\!\left(n,k\right)/(-1)^{{n-k}}$: notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); ¶ ‣ §26.1
(with $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind)
$\genfrac{[}{]}{0.0pt}{}{a_{1}+a_{2}+\dots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{{q}}$: $q$ -multinomial coefficient; §26.16
$\{\ldots\}$: 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
(with $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind)
$\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$: $6j$ symbol; (34.4.1)
$\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$: $9j$ symbol; (34.6.1)