Hi,
Represent f(x)=$ 2x^4 -12x^3 +24x^2 - 30x +9 $ and its successive differences in factorial notation
Solution:Let f(x) =$ 2x^4 - 12x^3 +24x^2 -30x +9 $
=$2x^{(2)} +bx^{(3)} +cx^{(2)} +dx +9 $
=2x(x-1)(x-2)(x-3) +bx(x-1)(x-2) +cx(x-1) +dx +9 where b,c and d are constants to be determined.
Putting x=1, we get,
-7=0+0+0+d+9
$ \therefore $ d=-16
Similarly, putting x=2 and x=3 in turn, we find c=2, b=0
f(x)=$ 2x^{(4)}+2x^{(2)} - 16x^{(1)} + 9 $
$ \Delta f(x)=8x^{(3)} + 4x -16 $
$ \Delta^2 f(x)=24x^{(2)}+ 4 $
$ \Delta^3 f(x)=48x $
$\Delta^4 f(x)= 48 $
Is this correct ?
Reply is appreciated.