 In this case the series expansion of this rational fraction is
where
is the binomial coefficient for formula_13 and is 0 otherwise.If 
the coefficient of formula_15 in formula_16 is thus

For formula_18 the term of index in this sum is a polynomial in of degree formula_19 with leading coefficient formula_20 This shows that there exists a unique polynomial formula_21 with rational coefficients which is equal to formula_22 for large enough.This polynomial is the Hilbert polynomial, and has the form

The least such that formula_24 for is called the Hilbert regularity.