structural:reformatting, information:filtering
Integer-valued polynomials

Each polynomial (t k) {\displaystyle {\tbinom {t}{k}}} is integer-valued: it has an integer value at all integer inputs t {\displaystyle t}. (One way to prove this is by induction on k using Pascal's identity.) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, (4) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring R of a characteristic 0 field K, a polynomial in K takes values in R at all integers if and only if it is an R -linear combination of binomial coefficient polynomials.