Lemma 43.15.4. Let $P$ be a polynomial of degree $r$ with leading coefficient $a$. Then
for any $t$.
Lemma 43.15.4. Let $P$ be a polynomial of degree $r$ with leading coefficient $a$. Then
for any $t$.
Proof. Let us write $\Delta $ the operator which to a polynomial $P$ associates the polynomial $\Delta (P) = P(t) - P(t - 1)$. We claim that
This is true for $r = 0, 1$ by inspection. Assume it is true for $r$. Then we compute
Thus the claim follows from the equality
The lemma follows from the fact that $\Delta (P)$ is of degree $r - 1$ with leading coefficient $ra$ if the degree of $P$ is $r$. $\square$
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