Month: October 2019

Introducing Operator Delta

/ Michael Xue / 2 Comments

The r^{th} order finite difference of functionf(x) is defined by

\Delta^r f(x) = \begin{cases} f(x+1)-f(x), r=1\\ \Delta(\Delta^{r-1}f(x)), r > 1\end{cases}

From this definition, we have

\Delta f(x) = \Delta^1 f(x) = f(x+1)-f(x)

and,

\Delta^2 f(x) = \Delta (\Delta^{2-1} f(x))

= \Delta (\Delta f(x))

= \Delta( f(x+1)-f(x))

= (f(x+2)-f(x+1)) - (f(x+1)-f(x))

= f(x+2)-2f(x)+f(x+1)

as well as

\Delta^3 f(x) = \Delta (\Delta^2 f(x))

= \Delta (f(x+2)-2f(x)+f(x+1))

= (f(x+3)-2f(x+1)+f(x+2)) - (f(x+2)-2f(x)+f(x+1))

= f(x+3)-3f(x+2)+3f(x+1)-f(x)

The function shown below generates \Delta^r f(x), r:1\rightarrow 5 (see Fig. 1).

delta_(g, n) := block(
    local(f),
    
    define(f[1](x), 
           g(x+1)-g(x)),

    for i : 2 thru n do (
        define(f[i](x), 
               f[i-1](x+1)-f[i-1](x))
    ),

    return(f[n])
);

Fig. 1

Compare to the result of expanding (f(x)-1)^r=\sum\limits_{i=0}^r(-1)^i \binom{r}{i} f(x)^{r-i}, r:1\rightarrow 5 (see Fig. 2)

Fig. 2

It seems that

\Delta^r f(x) = \sum\limits_{i=0}^r(-1)^i \binom{r}{i} f(x+r-i)\quad\quad\quad(1)

Lets prove it!

We have already shown that (1) is true for r= 1, 2, 3.

Assuming (1) is true when r=k-1 \ge 4:

\Delta^{k-1} f(x) = \sum\limits_{i=0}^{k-1}(-1)^i \binom{r}{i} f(x+k-1-i)\quad\quad\quad(2)

When r=k,

\Delta^k f(x) = \Delta(\Delta^{k-1} f(x))

\overset{(2)}{=}\Delta (\sum\limits_{i=0}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-1-i))

=\sum\limits_{i=0}^{k-1}(-1)^i \binom{k-1}{i}f(x+1+k-1-i)-\sum\limits_{i=0}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-1-i)

=(-1)^0 \binom{k-1}{0}f(x+k-0)

+\sum\limits_{i=1}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-i)-\sum\limits_{i=0}^{k-2}(-1)^i \binom{k-1}{i}f(x+k-1-i)

-(-1)^{k-1}\binom{k-1}{k-1}f(x+k-1-(k-1))

\overset{\binom{k-1}{0} = \binom{k-1}{k-1}=1}{=}

f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-i)-\sum\limits_{i=0}^{k-2}(-1)^i \binom{k-1}{i}f(x+k-1-i) -(-1)^{k-1}f(x)

=f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-i)+\sum\limits_{i=0}^{k-2}(-1)^{i+1}\binom{k-1}{i}f(x+k-1-i) -(-1)^{k-1}f(x)

\overset{j=i+1, i:0 \rightarrow k-2\implies j:1 \rightarrow k-1}{=}

f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i\binom{k-1}{i}f(x+k-i) + \sum\limits_{j=1}^{k-1}(-1)^j \binom{k-1}{j-1}f(x+k-j)-(-1)^{k-1}f(x)

= f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i\binom{k-1}{i}f(x+k-i) + \sum\limits_{i=1}^{k-1}(-1)^i\binom{k-1}{i-1}f(x+k-i)+(-1)^k f(x)

= f(x+k) + \sum\limits_{i=1}^{k-1}(-1)^i f(x+k-i) (\binom{k-1}{i} + \binom{k-1}{i-1})+(-1)^k f(x)

\overset{\binom{k-1}{i} + \binom{k-1}{i-1}=\binom{k}{i}}{=}

f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i \binom{k}{i} f(x+k-i)+(-1)^k f(x)

= (-1)^0 \binom{k}{0}f(x+k-0)+\sum\limits_{i=1}^{k-1}(-1)^i \binom{k}{i} f(x+k-i)+(-1)^k \binom{k}{k} f(x+k-k)

= \sum\limits_{i=0}^{k}(-1)^i \binom{k}{i} f(x+k-i)