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Assume we have sequence of polynomials : $P_i(q)$ - each term is polynomial in $q$. (With integer coefficients, but hopefully it is not important). We expect that there exists recurrence relation a ...

Example: How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this ...