I have a question on Rabinowitz and Davis: Methods of numerical Integration. They start to give a sequence for what they call The (composite) Integration Newton-Cotes formulas. This together with my problem is given in the 3 snippets below. I would like to know if their procedure and this procedure in general is unique: i.e. for a given $n$ equidistant points then $(2.5.11)$ with $(2.5.12)$ is the only way of choosing/defining the composite Newton-Cotes formulas: for $n=3$ we have Simpson's $\frac{3} {8}$ rule and for $n=4$ we have Boole's $\frac{2} {45}$ rule, etc. I would like to have that even the error will be unique in a sense, but can I expect that it will be literally unique that e.g. for the Simpson's rule it will necessarily be $-\frac{3}{80}h^5f^{(4)}\xi, a<\xi<b$, by the sole requirement that it has the order of exactness $3$: it integrates cubics exactly. And not $-\frac{3}{82}h^5f^{(4)}\xi, a<\xi<b$, say.
To reformulate this last goal of mine, can we really choose $c_k$'s and $d_k$'s in $(2.5.27)$ and $(2.5.28)$ resp. uniquely as some definite/fixed rational number for each $k$ which is best in some sense ? Could someone give a recursive/closed formulas for (the) $c_k$'s and $d_k$'s ? I cannot find it in the book.
