Digital Library of Mathematical Functions
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NIST

Notations

♦*♦ABCDEFGHIJKLMNOPQRSTUVWXYZ
!
n!: factorial; Common Notations and Definitions
!q
n!q: q-factorial; 5.18(i)
!!
n!!: double factorial; Common Notations and Definitions
ab: 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 for Fourier transforms; (1.14.5)
*
f*g: convolution product; (2.6.34)
×
G×H: Cartesian product of groups G and H; 23.1
×
a×b: vector cross product; (1.6.9)
implies; Common Notations and Definitions
is equivalent to; Common Notations and Definitions
/
S1/S2: set of all elements of S1 modulo elements of S2; 21.1
\
set subtraction; Common Notations and Definitions
asymptotic equality; (2.1.1)
backward difference operator; 3.10(iii)
del operator; (1.6.19)
2
Laplacian; 1.5(ii)
2
Laplacian for polar coordinates; 1.5(ii)
2
Laplacian for cylindrical coordinates; 1.5(ii)
2
Laplacian for spherical coordinates; 1.5(ii)
f
gradient of differentiable scalar function f; (1.6.20)
F
divergence of vector-valued function F; (1.6.21)
×F
curl of vector-valued function F; (1.6.22)
integral; 1.4(iv)
ab
Cauchy principal value; (1.4.24)
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
qx
q-integral; 17.2(v)
z¯
complex conjugate; (1.9.11)
|z|
modulus (or absolute value); (1.9.7)
a
magnitude of vector; (1.6.3)
xp
p-norm of a vector; 3.2(iii)
x2
Euclidean norm of a vector; 3.2(iii)
x
infinity (or maximum) norm of a vector; 3.2(iii)
Ap
p-norm of a matrix; 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)
nth q-derivative; 17.2(iv)
xn¯
falling factorial; 26.1
xn¯
rising factorial; 26.1
b0+a1b1+a2b2+
continued fraction; 1.12(i)
(a)n
Pochhammer’s symbol; 5.2(iii)
(z-1)!=Γ(z)
alternative notation; 5.1
(with Γ(z): gamma function)
(a,b)
open interval; Common Notations and Definitions
(a,b]
half-closed interval; Common Notations and Definitions
(n|P)
Jacobi symbol; 27.9
(n|p)
Legendre symbol; 27.9
(a;q)n
q-factorial (or q-shifted factorial); 17.2(i)
(a;q)n
q-factorial (or q-shifted factorial); 5.18(i)
(a;q)ν
q-shifted factorial (generalized); 17.2(i)
(a;q)
q-shifted factorial; 17.2(i)
(a1,a2,,ar;q)n
multiple q-shifted factorial; 17.2(i)
(a1,a2,,ar;q)
multiple q-shifted factorial; 17.2(i)
(a,z)!=γ(a+1,z)
notation used by Dingle (1973); 8.1
(with γ(a,z): incomplete gamma function)
(j1m1j2m2|j1j2j3-m3)
Clebsch–Gordan coefficient; 34.1
(mn)
binomial coefficient; (1.2.1)
(mn)
binomial coefficient; 26.3(i)
(n1+n2++nkn1,n2,,nk)
multinomial coefficient; 26.4(i)
(j1j2j3m1m2m3)
3j symbol; (34.2.4)
Λ,ϕ
distribution; 1.16(i)
f,ϕ
tempered distribution; (2.6.11)
δ,ϕ
Dirac delta distribution; 1.16(iii)
nk
Eulerian number; 26.14(i)
x
floor of x; Common Notations and Definitions
x
ceiling of x; Common Notations and Definitions
[z0,z1,,zn]
divided difference; 3.3(iii)
[a]κ
partitional shifted factorial; (35.4.1)
[a,b)
half-closed interval; Common Notations and Definitions
[a,b]
closed interval; Common Notations and Definitions
[p/q]f
Padé approximant; 3.11(iv)
[a,z]!=Γ(a+1,z)
notation used by Dingle (1973); 8.1
(with Γ(a,z): incomplete gamma function)
[nk]
Stirling cycle number; 26.13
[nm]q
q-binomial coefficient (or Gaussian polynomial); (17.2.27)
[nm]q
q-binomial coefficient (or Gaussian polynomial); 26.9(ii)
[nk]=s(n,k)/(-1)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
{}
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)