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Notations

!
factorial (as in n!); Common Notations and Definitions
!q
q-factorial (as in n!q); (5.18.2)
!!
double factorial; Common Notations and Definitions
𝐚𝐛: vector dot (or scalar) product; (1.6.2)
*
f*g: convolution product; (2.6.34)
×
𝐚×𝐛: vector cross product; (1.6.9)
×
G×H: Cartesian product of groups; §23.1
/
S1/S2: set of all elements of S1 modulo elements of S2; §21.1
set subtraction; Common Notations and Definitions
implies; Common Notations and Definitions
is equivalent to; Common Notations and Definitions
asymptotic equality; (2.1.1)
Poincaré asymptotic expansion; §2.1(iii)
backward difference operator; §3.10(iii)
del operator; (1.6.19)
2
Laplacian for spherical coordinates; §1.5(ii)
f
gradient of differentiable scalar function; (1.6.20)
𝐅
divergence of vector-valued function; (1.6.21)
×𝐅
curl of vector-valued function 𝐅; (1.6.22)
integral; §1.4(iv)
a(b+)
loop integral in : path begins at a, encircles b once in the positive sense, and returns to a.; §5.9(i)
P(1+,0+,1,0)
Pochhammer’s loop integral; §5.12
dqx
q-integral; §17.2(v)
ab
Cauchy principal value; (1.4.24)
f(c)
limit on left (or from below); (1.4.3)
f(c+)
limit on right (or from above); (1.4.1)
z¯
complex conjugate; (1.9.11)
xn¯
falling factorial; §26.1
xn¯
rising factorial; §26.1
|z|
modulus (or absolute value); (1.9.7)
𝐚
magnitude of vector; (1.6.3)
𝐱2
Euclidean norm of a vector; §3.2(iii)
𝐀p
p-norm of a matrix; §3.2(iii)
𝐱p
p-norm of a vector; §3.2(iii)
𝐱
infinity (or maximum) norm of a vector; §3.2(iii)
b0+a1b1+a2b2+
continued fraction; §1.12(i)
x
ceiling of x; Common Notations and Definitions
x
floor of x; Common Notations and Definitions
[z0,z1,,zn]f
divided difference; (3.3.34)
[a]κ
partitional shifted factorial; (35.4.1)
f[n](z)
nth q-derivative; §17.2(iv)
[a,b]
closed interval; Common Notations and Definitions
[a,b)
half-closed interval; Common Notations and Definitions
[a,z]!=Γ(a+1,z)
notation used by Dingle (1973); §8.1
(with Γ(a,z): incomplete gamma function)
[p/q]f
Padé approximant; §3.11(iv)
[nk]=(1)nks(n,k)
notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
(with s(n,k): Stirling number of the first kind)
[a1+a2++ana1,a2,,an]q
q-multinomial coefficient; §26.16
[nm]q
q-binomial coefficient (or Gaussian polynomial); (17.2.27)
(S)
cycle; §26.2
(z1)!=Γ(z)
alternative notation; §5.1
(with Γ(z): gamma function)
(a)n
Pochhammer’s symbol (or shifted factorial); §5.2(iii)
(a,b)
open interval; Common Notations and Definitions
(a,b]
half-closed interval; Common Notations and Definitions
(a,z)!=γ(a+1,z)
notation used by Dingle (1973); §8.1
(with γ(a,z): incomplete gamma function)
(m,n)
greatest common divisor (gcd); §27.1
(n|P)
Jacobi symbol; §27.9
(n|p)
Legendre symbol; §27.9
(a;q)n
q-Pochhammer symbol (or q-shifted factorial); §17.2(i)
(a1,a2,,ar;q)n
multiple q-Pochhammer symbol; §17.2(i)
(j1m1j2m2|j1j2j3m3)
Clebsch–Gordan coefficient; §34.1
(mn)
binomial coefficient; §1.2(i)
(n1+n2++nkn1,n2,,nk)
multinomial coefficient; §26.4(i)
(j1j2j3m1m2m3)
3j symbol; (34.2.4)
[nk]
Stirling cycle number of the first kind; (26.13.3)
{}
sequence, asymptotic sequence (or scale), or enumerable set; §2.1(v)
{z,ζ}
Schwarzian derivative; (1.13.20)
{nk}=S(n,k)
notation used by Knuth (1992), Graham et al. (1994), Rosen et al. (2000); §26.1
(with S(n,k): Stirling number of the second kind)
{j1j2j3l1l2l3}
6j symbol; (34.4.1)
{j11j12j13j21j22j23j31j32j33}
9j symbol; (34.6.1)
f,ϕ
tempered distribution; (2.6.11)
Λ,ϕ
inner-product of distribution; §1.16(i)
nk
Eulerian number; §26.14(i)