Introduction - Basic Difference Calculus - Discrete Fractional Calculus

Discrete Fractional Calculus (2015)

1. Basic Difference Calculus

Christopher Goodrich1 and Allan C. Peterson2

(1)

Department of Mathematics, Creighton Preparatory School, Omaha, NE, USA

(2)

Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE, USA

1.1. Introduction

In this section we introduce the basic delta calculus that will be useful for our later results. Frequently, the functions we consider will be defined on a set of the form

 $$\displaystyle{\mathbb{N}_{a}:=\{ a,a + 1,a + 2,\ldots \},}$$

where  $$a \in \mathbb{R},$$ or a set of the form

 $$\displaystyle{\mathbb{N}_{a}^{b}:=\{ a,a + 1,a + 2,\ldots,b\},}$$

where  $$a,b \in \mathbb{R}$$ and ba is a positive integer.

Definition 1.1.

Assume  $$f: \mathbb{N}_{a}^{b} \rightarrow \mathbb{R}$$ . If b > a, then we define the forward difference operator  $$\Delta $$ by

 $$\displaystyle{\Delta f(t):= f(t + 1) - f(t)}$$

for  $$t \in \mathbb{N}_{a}^{b-1}.$$

Note that in Definition 1.1 we make a slight abuse of notation by writing  $$\Delta f(t)$$ , as we shall do throughout this text. Technically, it would be more precise to write  $$(\Delta f)(t)$$ to emphasize that  $$\Delta f$$ is a function that is being evaluated at the point t. However, as long as one understands this true meaning of the notation, then we see no harm in using the simpler-to-read notation  $$\Delta f(t)$$ .

Definition 1.2.

We define the forward jump operator  $$\sigma$$ on  $$\mathbb{N}_{a}^{b-1}$$ by

 $$\displaystyle{\sigma (t) = t + 1.}$$

It is often convenient to use the notation  $$f^{\sigma }$$ to denote the function defined by the composition  $$f\circ \sigma,$$ that is

 $$\displaystyle{f^{\sigma }(t) = (f\circ \sigma )(t) = f(\sigma (t)) = f(t + 1),}$$

for  $$t \in \mathbb{N}_{a}^{b-1}.$$ Also, the operator  $$\Delta ^{n}$$ ,  $$n = 1,2,3,\ldots$$ is defined recursively by  $$\Delta ^{n}f(t) = \Delta (\Delta ^{n-1}f(t))$$ for  $$t \in \mathbb{N}_{a}^{b-n}$$ , where we assume the integer ba ≥ n. Finally,  $$\Delta ^{0}$$ denotes the identity operator, i.e.,  $$\Delta ^{0}f(t) = f(t).$$

In the following theorem we give several important properties of the forward difference operator.

Theorem 1.3.

Assume  $$f,g: \mathbb{N}_{a}^{b} \rightarrow \mathbb{R}$$ and  $$\alpha,\beta \in \mathbb{R}$$ , then for  $$t \in \mathbb{N}_{a}^{b-1}$$

(i)

 $$\Delta \alpha = 0;$$

(ii)

 $$\Delta \alpha f(t) =\alpha \Delta f(t);$$

(iii)

 $$\Delta \left [f + g\right ](t) = \Delta f(t) + \Delta g(t);$$

(iv)

 $$\Delta \alpha ^{t+\beta } = (\alpha -1)\alpha ^{t+\beta };$$

(v)

 $$\Delta \left [fg\right ](t) = f(\sigma (t))\Delta g(t) + \Delta f(t)g(t);$$

(vi)

 $$\Delta \left (\frac{f} {g}\right )(t) = \frac{g(t)\Delta f(t)-f(t)\Delta g(t)} {g(t)g(\sigma (t))},$$

where in (vi) we assume g(t) ≠ 0,  $$t \in \mathbb{N}_{a}^{b}.$$

Proof.

We will just prove (iv) and the quotient rule (vi). Since

 $$\displaystyle{\Delta \alpha ^{t+\beta } =\alpha ^{t+1+\beta } -\alpha ^{t+\beta } = (\alpha -1)\alpha ^{t+\beta }}$$

we have that (iv) holds. To see that the quotient rule (vi) holds, note that

 $$\displaystyle\begin{array}{rcl} \Delta \left (\frac{f} {g}\right )(t)& =& \frac{f(t + 1)} {g(t + 1)} -\frac{f(t)} {g(t)} {}\\ & =& \frac{f(t + 1)g(t) - f(t)g(t + 1)} {g(t)g(t + 1)} {}\\ & =& \frac{g(t)[f(t + 1) - f(t)] - f(t)[g(t + 1) - g(t)]} {g(t)g(\sigma (t))} {}\\ & =& \frac{g(t)\Delta f(t) - f(t)\Delta g(t)} {g(t)g(\sigma (t))}. {}\\ \end{array}$$

The proof of the product rule (v) is Exercise 1.2.  □ 

Due to the fact that (ii) and (iii) hold in Theorem 1.3 we say  $$\Delta $$ is a linear operator.

Next, we define the falling function.

Definition 1.4 (Falling Function).

For n a positive integer we define the falling function,  $$t^{\underline{n}}$$ , read t to the n falling, by

 $$\displaystyle{t^{\underline{n}}:= t(t - 1)(t - 2)\cdots (t - n + 1).}$$

Also we let t 0 : = 1. 

The falling function is defined so that the following power rule holds.

Theorem 1.5 (Power Rule).

The power rule

 $$\displaystyle{\Delta t^{\underline{n}} = nt^{\underline{n-1}},}$$

holds for n = 1,2,3,⋯ .

Proof.

Assume n is a positive integer and consider

 $$\displaystyle\begin{array}{rcl} \Delta t^{\underline{n}}& =& (t + 1)^{\underline{n}} - t^{\underline{n}} {}\\ & =& (t + 1)t(t - 1)\cdots (t - n + 2) - t(t - 1)(t - 2)\cdots (t - n + 1) {}\\ & =& t(t - 1)(t - 2)\cdots (t - n + 2)[(t + 1) - (t - n + 1)] {}\\ & =& nt^{\underline{n-1}}. {}\\ \end{array}$$

This completes the proof. □ 

A very important function in mathematics is the gamma function which is defined as follows.

Definition 1.6 (Gamma Function).

The gamma function is defined by

 $$\displaystyle{\Gamma (z) =\int _{ 0}^{\infty }e^{-t}t^{z-1}dt}$$

for those complex numbers z for which the real part of z is positive (it can be shown that the above improper integral converges for all such z).

Integrating by parts we get that

 $$\displaystyle\begin{array}{rcl} \Gamma (z + 1)& =& \int _{0}^{\infty }e^{-t}t^{z}dt {}\\ & =& [-e^{-t}t^{z}]_{ t\rightarrow 0+}^{t\rightarrow \infty }-\int _{ 0}^{\infty }(-e^{-t})zt^{z-1}dt {}\\ & =& z\Gamma (z) {}\\ \end{array}$$

when the real part of z is positive. We then use the very important formula

 $$\displaystyle\begin{array}{rcl} \Gamma (z + 1) = z\Gamma (z)& &{}\end{array}$$

(1.1)

to extend the domain of the gamma function to all complex numbers z ≠ 0, − 1, −2, ⋯ . Also note that since it can be shown that  $$\lim _{z\rightarrow 0}\vert \Gamma (z)\vert = \infty $$ it follows from (1.1) that

 $$\displaystyle{\lim _{z\rightarrow -n}\vert \Gamma (z)\vert = \infty,\quad n = 0,1,2,\ldots,}$$

which is a fundamental property of the gamma function which we will use from time to time. Another well-known important consequence of (1.1) is that

 $$\displaystyle{\Gamma (n + 1) = n!,\quad n = 0,1,2,\cdots \,.}$$

Because of this, the gamma function is known as a generalization of the factorial function.

Note that for n a positive integer

 $$\displaystyle\begin{array}{rcl} t^{\underline{n}}& =& t(t - 1)\cdots (t - n + 1) {}\\ & =& \frac{t(t - 1)\cdots (t - n + 1)\Gamma (t - n + 1)} {\Gamma (t - n + 1)} {}\\ & =& \frac{\Gamma (t + 1)} {\Gamma (t - n + 1)}. {}\\ \end{array}$$

Motivated by this above calculation, we extend the domain of the falling function in the following definition.

Definition 1.7.

The (generalized) falling function is defined by

 $$\displaystyle{t^{\underline{r}}:= \frac{\Gamma (t + 1)} {\Gamma (t - r + 1)}}$$

for those values of t and r such that the right-hand side of this equation makes sense. We then extend this definition by making the common convention that t r  = 0 when tr + 1 is a nonpositive integer and t + 1 is not a nonpositive integer. We also use the convention given in Oldham and Spanier [152, equation (1.3.4)] that

 $$\displaystyle{ \frac{\Gamma (-n)} {\Gamma (-N)} = (-n - 1)^{\underline{N-n}} = (-1)^{N-n}\frac{N!} {n!},}$$

where n and N are nonnegative integers.

The motivation for the first convention in Definition 1.7 is that whenever tr + 1 is a nonpositive integer and t + 1 is not a nonpositive integer, then

 $$\displaystyle{\lim _{s\rightarrow t}s^{\underline{r}} =\lim _{ s\rightarrow t} \frac{\Gamma (s + 1)} {\Gamma (s - r + 1)} = 0.}$$

A similar remark motivates the second convention mentioned in Definition 1.7. Whenever these conventions are used one should always verify the conclusion by taking appropriate limits. This step will usually not be included in our calculations.

Next we state and prove the generalized power rules.

Theorem 1.8 (Power Rules).

The following (generalized) power rules

 $$\displaystyle\begin{array}{rcl} \Delta (t+\alpha )^{\underline{r}} = r(t+\alpha )^{\underline{r-1}},& &{}\end{array}$$

(1.2)

and

 $$\displaystyle\begin{array}{rcl} \Delta (\alpha -t)^{\underline{r}} = -r(\alpha -\sigma (t))^{\underline{r-1}},& &{}\end{array}$$

(1.3)

hold, whenever the expressions in these two formulas are well defined.

Proof.

Consider

 $$\displaystyle\begin{array}{rcl} \Delta (t+\alpha )^{\underline{r}}& =& (t +\alpha +1)^{\underline{r}} - (t+\alpha )^{\underline{r}} {}\\ & =& \frac{\Gamma (t +\alpha +2)} {\Gamma (t +\alpha +2 - r)} - \frac{\Gamma (t +\alpha +1)} {\Gamma (t +\alpha +1 - r)} {}\\ & =& \frac{[(t +\alpha +1) - (t +\alpha +1 - r)]\Gamma (t +\alpha +1)} {\Gamma (t +\alpha +2 - r)} {}\\ & =& r \frac{\Gamma (t +\alpha +1)} {\Gamma (t +\alpha -r + 2)} {}\\ & =& r(t+\alpha )^{\underline{r-1}}. {}\\ \end{array}$$

Hence (1.2) holds.

To see that (1.3) holds, consider

 $$\displaystyle\begin{array}{rcl} \Delta (\alpha -t)^{\underline{r}}& =& \Delta \left ( \frac{\Gamma (\alpha -t + 1)} {\Gamma (\alpha -t + 1 - r)}\right ) {}\\ & =& \frac{\Gamma (\alpha -t)} {\Gamma (\alpha -t - r)} - \frac{\Gamma (\alpha -t + 1)} {\Gamma (\alpha -t + 1 - r)} {}\\ & =& [(\alpha -t - r) - (\alpha -t)] \frac{\Gamma (\alpha -t)} {\Gamma (\alpha -t + 1 - r)} {}\\ & =& -r \frac{\Gamma (\alpha -t)} {\Gamma (\alpha -t + 1 - r)} {}\\ & =& -r(\alpha -\sigma (t))^{\underline{r-1}}. {}\\ \end{array}$$

Hence the power rule (1.3) holds. □ 

Note that when n ≥ k ≥ 0 are integers, then the binomial coefficient satisfies

 $$\displaystyle{\binom{n}{k}:= \frac{n!} {(n - k)!k!} = \frac{n(n - 1)\cdots (n - k + 1)} {k!} = \frac{n^{\underline{k}}} {\Gamma (k + 1)}.}$$

Motivated by this we next define the (generalized) binomial coefficient as follows.

Definition 1.9.

The (generalized) binomial coefficient  $$\binom{t}{r}$$ is defined by

 $$\displaystyle{\binom{t}{r}:= \frac{t^{\underline{r}}} {\Gamma (r + 1)}}$$

for those values of t and r so that the right-hand side is well defined. Here we also use the convention that if the denominator is undefined, but the numerator is defined, then  $$\binom{n}{k} = 0.$$

Theorem 1.10.

The following hold

(i)

 $$\Delta \binom{t}{r} = \binom{t}{r - 1};$$

(ii)

 $$\Delta \binom{r + t}{t} = \binom{r + t}{t + 1};$$

(iii)

 $$\Delta \Gamma (t) = (t - 1)\Gamma (t),$$

whenever these expressions make sense.

The proof of this theorem is left as an exercise (Exercise 1.13).