The Difference Operator

We will now look at a rather nifty operator known as the difference operator on a real-valued function $f$ which can be seen as an analogue to the derivative of $f$.

Definition: Let $f$ be a real-valued function. Then the Difference Operator $\Delta$ on $f$ is $\Delta f(n) = f(n + 1) - f(n)$.

For example, if $f(n) = n^3 - 2n$ then:

(1)

\begin{align} \quad \Delta f(n) = f(n + 1) - f(n) \\ \quad \Delta f(n) = \left [ (n + 1)^3 - 2(n + 1) \right ] - \left [ n^3 - 2n \right ] \\ \quad \Delta f(n) = n^3 + 3n^2 + 3n + 1 - 2n - 2 - n^3 + 2n \\ \quad \Delta f(n) = 3n^2 + 3n -1 \end{align}

One important property is that $\Delta$ is linear - that is, for $f$ and $g$ as real-valued functions and $a \in \mathbb{R}$, $\Delta$ satisfies the additivity property $\Delta (f + g) = \Delta f + \Delta g$ and the homogeneity property $\Delta (af) = a \Delta f$ as proven in the following theorems.

Theorem 1: If $f$ and $g$ are real-valued functions then $\Delta (f + g) = \Delta f + \Delta g$.

• Proof: The proof of Theorem 1 is straightforward:

(2)

\begin{align} \quad \Delta(f + g) = [f(n + 1) + g(n + 1)] - [f(n) + g(n)] = [f(n+ 1) - f(n)] + [g(n+1) - g(n)] = \Delta f + \Delta g \quad \blacksquare \end{align}

Theorem 2: If $a \in \mathbb{R}$ and $f$ is a real-valued function then $\Delta (af) = a \Delta f$.

• Proof: Like Theorem 1, the proof of Theorem 2 is straightforward:

(3)

\begin{align} \quad \Delta (af) = af(n + 1) - af(n) = a[f(n+1) - f(n)] = a \Delta f \quad \blacksquare \end{align}

Furthermore, we have that if $f(n) = c$ where $c$ is a constant then $\Delta f = \Delta c = 0$ as we prove below.

Theorem 3: If $c \in \mathbb{R}$ and $f(n) = c$ then $\Delta f = 0$.

• Proof:

(4)

\begin{align} \quad \Delta f = \Delta c = c - c = 0 \quad \blacksquare \end{align}