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DLMF
Index
Notations
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Index Z
Notations A
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
♦
𝐀
−
1
matrix inverse;
§1.2(vi)
𝐀
m by n matrix;
§1.2(v)
!
factorial (as in
n
!
);
Common Notations and Definitions
!
q
q
-factorial (as in
n
!
q
);
(5.18.2)
!!
double factorial;
Common Notations and Definitions
⋅
𝐚
⋅
𝐛
: 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
/
S
1
/
S
2
: set of all elements of
S
1
modulo elements of
S
2
;
§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
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)
∫
a
(
b
+
)
loop integral in
ℂ
: 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
∫
⋯
d
q
x
q
-integral;
§17.2(v)
⨍
a
b
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)
x
n
¯
falling factorial;
§26.1
x
n
¯
rising factorial;
§26.1
|
z
|
modulus (or absolute value);
(1.9.7)
‖
𝐚
‖
magnitude of vector;
(1.6.3)
‖
𝐱
‖
2
Euclidean norm of a vector;
§3.2(iii)
‖
𝐀
‖
p
p
-norm of a matrix;
§3.2(iii)
‖
𝐱
‖
p
p
-norm of a vector;
§3.2(iii)
‖
𝐱
‖
∞
infinity (or maximum) norm of a vector;
§3.2(iii)
b
0
+
a
1
b
1
+
a
2
b
2
+
⋯
continued fraction;
§1.12(i)
⌈
x
⌉
ceiling of
x
;
Common Notations and Definitions
⌊
x
⌋
floor of
x
;
Common Notations and Definitions
[
z
0
,
z
1
,
…
,
z
n
]
f
divided difference;
(3.3.34)
[
a
]
κ
partitional shifted factorial;
(35.4.1)
f
[
n
]
(
z
)
n
th
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)
[
n
k
]
=
(
−
1
)
n
−
k
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 first kind
)
[
a
1
+
a
2
+
⋯
+
a
n
a
1
,
a
2
,
…
,
a
n
]
q
q
-multinomial coefficient;
§26.16
[
n
m
]
q
q
-binomial coefficient (or Gaussian polynomial);
(17.2.27)
(
S
)
cycle;
§26.2
(
z
−
1
)
!
=
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)
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
multiple
q
-Pochhammer symbol;
§17.2(i)
(
j
1
m
1
j
2
m
2
|
j
1
j
2
j
3
m
3
)
Clebsch–Gordan coefficient;
§34.1
𝐀
∗
adjoint of matrix;
§1.3(iv)
𝐀
H
Hermitian conjugate of matrix;
(1.2.30)
𝐀
T
transpose of matrix;
(1.2.28)
(
m
n
)
binomial coefficient;
§1.2(i)
(
n
1
+
n
2
+
⋯
+
n
k
n
1
,
n
2
,
…
,
n
k
)
multinomial coefficient;
§26.4(i)
(
j
1
j
2
j
3
m
1
m
2
m
3
)
3
j
symbol;
(34.2.4)
[
𝐀
,
𝐁
]
commutator;
(1.2.66)
[
n
k
]
Stirling cycle number of the first kind;
(26.13.3)
⟨
f
,
g
⟩
inner product over functions;
(1.18.12)
⟨
𝐮
,
𝐯
⟩
inner product over vectors;
(1.2.40)
‖
𝐯
‖
vector norm (
l
²);
(1.2.46)
{
…
}
sequence, asymptotic sequence (or scale), or enumerable set;
§2.1(v)
{
z
,
ζ
}
Schwarzian derivative;
(1.13.20)
{
n
k
}
=
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
)
{
j
1
j
2
j
3
l
1
l
2
l
3
}
6
j
symbol;
(34.4.1)
{
j
11
j
12
j
13
j
21
j
22
j
23
j
31
j
32
j
33
}
9
j
symbol;
(34.6.1)
⟨
f
,
ϕ
⟩
tempered distribution;
(2.6.11)
⟨
Λ
,
ϕ
⟩
action of distribution on test function;
§1.16(i)
⟨
n
k
⟩
Eulerian number;
§26.14(i)