Proofs of central-difference interpolation formulas

Abstract

Let D and E denote the differentiation and forward-shift operators, respectively. A linear operator on the space of polynomials is called shift-invariant if it commutes with the operator E. Each shift-invariant operator can be expressed as a power series in D, C,“=O c,D”. A delta operator is a shiftinvariant operator with c0 = 0 and c, # 0. For each delta operator Q there is a unique sequence of polynomials p&), p,(x), p*(x),..., where p,(x) is a polynomial of degree n, J+,(X)1, and O=pi(O) =p*(O)= ..a , such that QPJx) = np,-l(x), n = 1,2,3,.... These polynomials have been called poweroids [ 10, p. 3351, basic polynomials [8, p. 592; 3, p. 1811 and associated polynomials [7, p. 105; 5, sect. 51. The principal tool for our proofs is the following generalization of the Taylor expansion formula.