Integer-valued Polynomial - Classification

Classification

In fact integer-valued polynomials can be described fully. Inside the polynomial ring Q of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials

P_k(t) = t(t − 1)...(t − k + 1)/k!

for k = 0,1,2, ..., i.e., the binomial coefficients. In other words, every integer-valued polynomial can be written as an integer linear combination of binomial coefficients in exactly one way. The proof is by the method of discrete Taylor series: binomial coefficients are integer-valued polynomials, and conversely, the discrete difference of an integer series is an integer series, so the discrete Taylor series of an integer series generated by a polynomial has integer coefficients (and is a finite series).