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It is well known that the polynomials, which together with their derivatives have integral values at a fixed point x = xo, have the basis (x -Xo) n/n!, n = O, 1, 2,* * over the integers. That is to say every polynomial with that property is of the form E ck(xCxo)k/k! where co, cl, , cn are integers. Similarly it is known that a basis over the integers for those polynomials whose values are integral at all integers is given by 1, x(x1) *.. (x-n + 1)/n!, n = 1, 2, It is the purpose of this note to construct a basis over the integers for the polynomials which together with all their derivatives have integral values at all integers. We shall call such polynomials Ipolynomials. In order to construct this basis we prove a few introductory lemmas:

On the polynomials whose derivatives have integral values at the integers