NEWTON BASIS RELATIONS AND APPLICATIONS TO INTEGER-VALUED POLYNOMIALS AND q-BINOMIAL COEFFICIENTS

Let K be a field. Generalizing the binomial coefficient polynomials ( X n ) , the Newton interpolation basis polynomials relative to an infinite sequence a = (ai)i=0 of distinct elements of K are defined by [X n ] a = ∏n−1 i=0 X−ai an−ai ∈ K[X ]. There is a complete and natural set of relations for these polynomials, namely, [X m ] a [X n ] a = ∑m+n l=max(m,n)[m,n, l]a [X l ] a , where the coefficients [m,n, l]a are in K for all m,n, l. We derive formulas for the [m,n, l]a that uniquely characterize the linear iterative sequences and the sequences of squares and triangular numbers and are expressed, respectively, in terms of the q-multinomial coefficients and the ordinary multinomial coefficients. We also use these relations to find a D-algebra presentation of the ring Int(S,D) = {f ∈ K[X ] : f(S) ⊆ D} of integer-valued polynomials on S in D, where D is any integral domain with quotient field K and S is any subset of D such that an ∈ S and [X n ] a ∈ Int(S,D) for all n.