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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?

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The denominator of the transfer function of an analog Butterworth filter is a (non-normalized) Butterworth polynomial. Also the numerator is just a constant (i.e. the coefficients of higher order are 0).

You are designing a discrete filter here, so this does no longer apply. Your numerator has three non-zero coefficients, and the denominator is not a Butterworth polynomial.

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.

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