Tabular reconstruction of differences from a curtailed set of moments is ordinarily impossible because of a gap on the integral side of the difference table. This gap can be closed by substituting an associated polynomial for the one that is being operated upon. The associated polynomial has, at the end of a contracted range, terminal differences and moments individually proportional to corresponding differences and moments of the polynomial from which it was derived. The function is applied to the problem of matching moments to form least squares approximations.
"An Associated Polynomial for Least Squares Approximations." Ann. Math. Statist. 31 (4) 969 - 981, December, 1960. https://doi.org/10.1214/aoms/1177705670