# Butterworth Polynomial from Butterworth Filter

For the butterworth filter, in Matlab command:

[B,A]=butter(2,0.5,'low')


I find this result:

B = [0.29289  0.58579  0.29289]
A = [1  -3.33067e-16  0.17157]


From here I cannot reach the Normalized Butterworth polynomial:

G = [1  1.4142  1]


Nor the poles:

P = [-0.7071+i0.7071  -0.7071-i0.7071]


Which is the relation or the proper parameter i am missing, if any?

You can check that if you design an analog Butterworth adding the parameter 's' when calling the function, you will indeed get a one non-zero element array B and the coefficients of a non-normalized Butterworth polynomial in A.