# More Properties of the Difference Operator

Recall from The Difference Operator page that if $f$ is a real-valued function then the difference operator $\Delta$ on $f$ is defined to be:

(1)We proved some very basic properties of the difference operator. We showed that if $f$ and $g$ are real-valued functions that $\Delta$ satisfies the additivity property, that is $\Delta (f + g) = \Delta f + \Delta g$. We also showed that if $f$ is a real-valued function and $a \in \mathbb{R}$ then $\Delta (af) = a \Delta f$ and so $\Delta$ satisfies the homogeneity property. Furthermore, we showed that if $f$ is a constant function then $\Delta f = 0$.

So far, we've noted that $\Delta$ behaves very much like the differentiation operator $D$ on real-valued functions. Recall that for any real-valued function $f$ that provided the limit exists:

(2)By setting $h = 1$ and replacing the variable $x$ with $n$ we get the difference operator $\Delta$, so it's somewhat natural that $\Delta$ inherits many of the properties that the differentiation operator $D$ has. We will look at some more of these properties in the following theorems, the first of which gives us an analogue to the differentiation rule that $D(x^n) = nx^{n-1}$.

Theorem 1: If $f(x) = x^{\underline{n}} = x \cdot ( x - 1) \cdot ... \cdot (x - n + 1)$ then $\Delta f(x) = nx^{\underline{n-1}}$. |

*Recall that $x^{\underline{n}}$ is simply a Falling Factorial.*

**Proof:**

Recall from calculus that if $f(x) = e^x$ then $\frac{df}{dx} = e^x$. We would like to find a function $f$ such that $\Delta f = f$. Fortunately one such function exists as proven in Theorem 2.

Theorem 2: If $f(x) = 2^x$ then $\Delta f(x) = \Delta 2^x = 2^x$. |

**Proof:**

Recall from The Generalized Binomial Coefficient Formula page that the generalized binomial coefficient for $n \in \mathbb{R}$ and $k \in \{0, 1, 2, ... \}$ is given by:

(5)We will derive a rather interesting equality with regards to the difference of generalized binomial coefficients.

Theorem 3: If $f(x) = \binom{x}{k}$ then $\Delta f(x) = \binom{x}{k - 1}$. |

**Proof:**We use the homogeneity property of $\Delta$ as well as Theorem 1 to arrive at: