Concrete Mathematics(2nd Edition) - A Note on Notation

$\ln x$
natural logarithm: $\log_e x$
$\lg x$
binary logarithm: $\log_2 x$
$\log x$
common logarithm: $\log_{10} x$
$\lfloor x\rfloor$
floor: $\max\{n\ |\ n \le x, \ \text {integer $n$}\}$
$\lceil x \rceil$
ceiling: $\min \{n \ |\ n \ge x, \ \text {integer $n$} \}$
$x \mod y$
remainder: $x-y \lfloor x/y \rfloor$
$\{ x\}$
fractional part: $x \mod 1$
$\sum {f(x)\delta x}$
indefinite summation
$\sum_a^b {f(x) \delta x}$
definite summation
$x^{\underline n}$
falling factorial power: $x!/(x-n)!$
$x^{\overline n}$
rising factorial power: $\Gamma(x+n)/\Gamma(x)$
$n_i$
subfactorial: $n!/0!-n!/1!+\cdots+(-1)^nn!/n!$
$\Re_z$
real part: $x, \ \text{if $z = x + iy$}$
$\Im_z$
imaginary part: $y, \ \text{if $z = x+ iy$}$
$H_n$
harmonic number: $1/1+\cdots+1/n$
$H_n^{(x)}$
generalized harmonic number: $1/1^x+\cdots+1/n^x$
$f^{(m)}(z)$
$m$th derivative of $f$ at $z$
$\begin{bmatrix}n \\m \end{bmatrix}$
Stirling cycle number (the “first kind”)
$\begin{Bmatrix}n \\m \end{Bmatrix}$
Striling cycle number (the “second kind”)
$\left\langle \begin{matrix} n \\ m \end{matrix} \right\rangle$
Eulerian number
$\left\langle \left\langle \begin{matrix} n \\ m \end{matrix} \right\rangle \right\rangle$
Second order Eulerian number
$(a_m,\dots,a_0)_b$
radix notation for $\sum_{k=0}^ma_kb^k$
$K(a_1,\dots,a_n)$
continuant polynomial
$F\left(\begin{array}{c} a,b \\ c \end{array} \big | z\right)$
hypergeometric function
$\#A$
cardinality: number of elements in the set A
$[z^n]f(z)$
coefficient of $z^n$ in $f(z)$
$[\alpha..\beta]$
closed interval: the set $\{x \ | \ \alpha \le x \le \beta\}$
$[m=n]$
$1$ if $m=n$, otherwise $0^*$
$[m \setminus n]$
$1$ if $m$ divides $n$, otherwise $0^*$
$[m \setminus \setminus n]$
$1$ if $m$ exactly divides $n$, otherwise $0^*$
$[m \bot n]$
$1$ if $m$ is relatively prime to $n$, otherwise $0^*$