Exponential functions, in a general sense, are defined as

So for $j=0$, this is the regular exponential.

The main application is to apply that exponential function to a matrix.

Here is a minimal example:

from padexp import Exponential
import numpy as np
e = Exponential(4) # to compute the functions E_j for 0 ≤ j ≤ 4
M = np.array([[1.,2.],[3.,4]])
e(M) # returns a list containing [E_0(M), E_1(M),...,E_4(M)]

This code is useful for exponential integrators, and is a port of the expint Matlab package.