

In [8]:

    
import numpy as np
import sympy as sy
from sympy import init_printing
init_printing()
import control.matlab as cm

The Fibonacci sequence

\begin{equation} y(k+2) - y(k+1) - y(k) = 0 \end{equation}



In [20]:

    
z = sy.symbols('z')
y0, y1 = sy.symbols('y0, y1', real=True)



In [21]:

    
den = z*z - z -1
num = z*z
Y = num/sy.factor(den)
sy.factor(den)









    Out[21]:





$$z^{2} - z - 1$$



In [22]:

    
sy.factor(Y)









    Out[22]:





$$\frac{z^{2}}{z^{2} - z - 1}$$



In [23]:

    
sy.apart(Y)









    Out[23]:





$$\frac{z + 1}{z^{2} - z - 1} + 1$$



In [24]:

    
sy.apart( z/den)









    Out[24]:





$$\frac{z}{z^{2} - z - 1}$$



In [25]:

    
p1 = (1 + sy.sqrt(5))/2
p2 = (1 - sy.sqrt(5))/2
sy.expand((z-p1)*(z-p2))









    Out[25]:





$$z^{2} - z - 1$$



In [28]:

    
Y = z/( (z-p1)*(z-p2))
Y









    Out[28]:





$$\frac{z}{\left(z - \frac{1}{2} + \frac{\sqrt{5}}{2}\right) \left(z - \frac{\sqrt{5}}{2} - \frac{1}{2}\right)}$$



In [29]:

    
sy.apart(Y)









    Out[29]:





$$\frac{z}{z^{2} - z - 1}$$



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