The Nash cascade is a sequence of $N$ linear stores in series, each having a residence time of $K$ time units. The resulting mathematical form of the Nash-cascade is:

$$h(t) = (t/K)^{N-1}*exp\left(\frac{-t/K}{K*\Gamma(N)}\right)$$

where $h(t)$ is the unit hydrograph in terms of a unit discharge and $\Gamma$ is the gamma function $\gamma(N)=(N-1)!$ for integer values $N$. $N$ can also have rational values.

|Nash.py

from pylab import *
from scipy import *
from math import *
from numpy import *
 
N = 5
k = 0.5
t = arange(0.0, 60, 1)
 
def Nash(t,N,k):
    h = (t/k)**(N-1)*exp((-t/k)/(k*gamma(N)))
    return h
 
plot(t, Nash(t,N,k))
xtext = xlabel('time')
ytext = ylabel('response')
setp(xtext, size='medium', name='courier', weight='bold', color='g')
setp(ytext, size='medium', name='helvetica', weight='light', color='b')
show()