Alternating Exponential Generating Functions

List of exponential generating functions (egf) often used in the resolution of divergent series.

$x \in \mathbb{C}$ denotes the rational, real or complex variable.
$a \in \mathbb{C}$ denotes a rational, real or complex constant.

$j \in \mathbb{N}$ denotes the number of (equitable) spacing between all the terms of the series.
$k \in \mathbb{N}$ denotes the number of shifts (leading zeros in front) of the series.

$A_n(k)$ denotes the Eulerian number $A(n,k)$ .
$B_n$ denotes the n-th Bernoulli number.
$B_n^{+}$ denotes the n-th Bernoulli number in the context of $B_1^{+} = \frac{1}{2}$ .
$B_n^{-}$ denotes the n-th Bernoulli number in the context of $B_1^{-} = -\frac{1}{2}$ .
$E_n$ denotes the n-th Euler number.

$\zeta(x)$ denotes the Riemann zeta function.
$\zeta(x,a)$ denotes the Hurwitz zeta function.
$\eta(x)$ denotes the Dirichlet eta function.
$\Gamma(x)$ denotes the Gamma function.
$\psi^{(0)}(x) \,=\, \frac{d}{dx} \ln\Gamma(x)$ denotes the digamma function.
$\psi^{(1)}(x) \,=\, \frac{d}{dx}\psi^{(0)}(x) \,=\, \frac{d^2}{dx^2} \ln\Gamma(x) \,=\, \zeta(2,x)$ denotes the trigamma function.