Numerical methods: forward difference operator

Question

$${ \text{If }y = \frac{1}{{(4 x + 1)} {(4 x + 5)} {(4 x + 9)}}, }\\ { \text{find } z = \Delta ^ {2} y .}$$ Can someone help with this? I tried it using factorial notation but apparently my answer is partially correct. Any help would be much appreciated.

Answer 1

By induction, for $a,b\in \mathbb{R}$ with $a\ne 0$ and denoting forward difference by $\Delta$, we have $$ {\Delta}^n (ax+b)^{-1} = (-1)^n\cdot n! \cdot a^n \cdot \prod_{k=0}^{n} \left(ax+b +ak\right)^{-1} $$So, your question can be rewritten as $$ z = \Delta^2 y = \frac{1}{2!\cdot 4^2}\Delta^4 (4x+1)^{-1} = \frac{4!\cdot 4^4}{2!\cdot 4^2}\cdot \prod_{k=0}^{4} \left(4x+1 +4k\right)^{-1} $$ $$ = \frac{192}{(4x+1)(4x+5)(4x+9)(4x+13)(4x+17)} $$