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Mixed finite difference of a polynomial

If

p

has degree

n

and leading coefficient

a_{n}

, then for nonzero steps

r_{1}, \dots, r_{n}

,

D_{r_{1}} \dots D_{r_{n}} p = n! a_{n} \prod_{i = 1}^{n} r_{i} .

Consequently a degree-

n

polynomial cannot be the sum of

n

functions each periodic with some nonzero period.

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