Would you elaborate a bit more on what your question is?
Are you wanting code to simplify the first element of p2 and p3 to be the second element of p2 and p3, respectively?
Also, by "Gauss' formula" do you mean what's mentioned in the following: https://math.stackexchange.com/questions/2852717/understanding-gausss-product-formula-as-cited-in-golombs-shift-register-sequen.
It appears that p3[x] is just p2[x, Infinity] so only the code for p2 seems necessary.
