Suppose is a polynomial of degree . Prove that
by induction on . Here, is the derivative of , and is to be interpreted as .
A hint I was provided was to use integration by parts.
I'll let you the case . Now, we'll suposse that the proposition is true for some poyinomial of degree . We have to prove that this proposition is true for . Let be a polynomial of degree .
Integrating by parts
But the proposition is true for the polynomial of degree k, then:
And now, conclude