Definition of $\Delta$ .

The derivative of a function $f\colon\mathbb{R}\to\mathbb{R}$ is defined to be the expression $$\frac{df}{dx}:=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h},$$ which makes sense whenever $f$ is differentiable (at least at $x$ ). However, the expression $$\frac{f(x+h)-f(x)}{h}$$ makes sense even without $f$ being continuous, as long as $h\neq 0$ . The expression is called a finite difference. The simplest case when $h=1$ , written $$\Delta f(x):=f(x+1)-f(x),$$ is called the forward difference of $f$ . For other non-zero $h$ , we write $$\Delta_h f(x):=\frac{f(x+h)-f(x)}{h}.$$ When $h=-1$ , it is called a backward difference of $f$ , sometimes written $\nabla f(x):=\Delta_{-1} f(x)$ . Given a function $f(x)$ and a real number $h\neq 0$ , if we define $y=\frac{x}{h}$ and $g(y)=\frac{f(hy)}{h}$ , then we have $$\Delta g(y)=\Delta_h f(x).$$ Conversely, given $g(y)$ and $h\neq 0$ , we can find $f(x)$ such that $\Delta g(y)=\Delta_h f(x)$ .

Some Properties of $\Delta$ .

It is easy to see that the forward difference operator $\Delta$ is linear:

1. $\Delta(f+g)=\Delta(f)+\Delta(g)$
2. $\Delta(cf)=c\Delta(f)$ , where $c\in\mathbb{R}$ is a constant.

1. $\Delta(c)=0$ for any real-valued constant function $c$ , and
2. $\Delta(I)=1$ for the identity function $I(x)=x$ . constant.

1. $\Delta a^x=(a-1)a^x$ for any real number $a$
2. $\Delta x^{(n)}=nx^{(n-1)}$ where $x^{(n)}$ denotes the falling factorial polynomial
3. $\Delta b_n(x)=nx^{n-1}$ , where $b_n(x)$ is the Bernoulli polynomial of order $n$ .

From $\Delta$ , we can also form other operators. For example, we can iteratively define \begin{eqnarray} &&\Delta^{1}f:=\Delta f \\ &&\Delta^{k}f:=\Delta(\Delta^{k-1}f),\quad\mbox{where }k>1. \end{eqnarray}Of course, all of the above can be readily generalized to $\Delta_h$ . It is possible to show that $\Delta_h f$ can be written as a linear combination of $$\Delta f,\Delta^2 f,\ldots,\Delta^h f.$$

Difference Equation.

Suppose $F\colon\mathbb{R}^n\to\mathbb{R}$ is a real-valued function whose domain is the $n$ -dimensional Euclidean space. A difference equation (in one variable $x$ ) is the equation of the form $$F(x,\Delta_{h_1}^{k_1}f,\Delta_{h_2}^{k_2}f,\ldots,\Delta_{h_n}^{k_n}f)=0,$$ where $f:=f(x)$ is a one-dimensional real-valued function of $x$ . When $h_i$ are all integers, the expression on the left hand side of the difference equation can be re-written and simplified as $$G(x,f,\Delta f,\Delta^{2}f,\ldots,\Delta^{m}f)=0.$$ Difference equations are used in many problems in the real world, one example being in the study of traffic flow.