Differences and derivatives are not the same, though there is the obvious analogy. If you want to take derivatives and antiderivatives, you want to write in the x^k basis or the x^k/k! basis. If you want to take differences and sums, you want to write in the falling factorial basis or the x choose k basis.
If you get a non constant, yes. For a linear function, f(a+1) - f(a) = f’(a). Inductively you can then show that the nth one-step difference of a degree n polynomial f at a point a is f^(n)(a). But this doesn’t work for anything but n. Thanks for pointing that out!
Differences and derivatives are not the same, though there is the obvious analogy. If you want to take derivatives and antiderivatives, you want to write in the x^k basis or the x^k/k! basis. If you want to take differences and sums, you want to write in the falling factorial basis or the x choose k basis.
If you get a non constant, yes. For a linear function, f(a+1) - f(a) = f’(a). Inductively you can then show that the nth one-step difference of a degree n polynomial f at a point a is f^(n)(a). But this doesn’t work for anything but n. Thanks for pointing that out!
Ah, yes, that’s a good point, because the leading coefficient be the same whether you use the x^k basis or the falling factorial basis.