I. INTRODUCTION
A perfect power is a number which has a rational root. Chase . An integral perfect power is the irrational root that is an integer. A finite difference is a mathematical expression of the form, and a forward difference is of the form .
In this article, an examination on the set of even numbers: and the set of odd numbers: reveals that the first difference of the square of consecutive elements of both the set of even and odd numbers is even. The second difference of the two distinct sets is , a singleton set.
Clearly, the third difference is . The emerging pattern motivates the investigation of whether or not this pattern persists for all perfect squares of elements of even and odd sets, resulting from integers. In their contribution in Exponential Diophantine Equations, Shorey and Tijdeman investigated perfect powers at integral values of a polynomial with rational integer coefficients and obtain in particular the following.
Let be a polynomial with rational coefficients and with at least two simple rational zeros.
Suppose b \neq 0, m \geq 0, x and y with |y| > 1 are rational integers. Then the equation f(x) = by^{m} implies that m is bounded by a computable number depending only on b and f. Also, in their contribution in the difference of perfect powers of integers, Ladan, Ukwu and Apine^{[4]} investigated order of integral perfect powers and proved that "if any number of consecutive integers are raised to a positive integral power k, then the k^{th} difference is equal to k!"
More generally, the question at the heart of the matter is the following. What is the computational disposition of the orders of difference of integral perfect powers of successive elements of set of even and odd numbers? The Review of Literature shows that no such investigation has been undertaken. Thus, this article adds to the existing body of knowledge, by providing answers to the above question.
II. METHODS
2.1 Preliminary Definitions
In what follows, the difference of finite order will be defined.
2.1.1 Difference of Order One (1)
Given a sequence , define the difference of order one at j with respect to the sequence by:
\[\Delta\left(U_{j}\right) = U_{j+1} - U_{j}, \text{for every integral} j.\]
2.1.2 Higher Order Difference
Higher difference can be defined recursively by:
\[\Delta^{k}(U_{j}) = \Delta \Delta^{k-1}(U_{j}) = \Delta^{k-1} \Delta(U_{j}) = \Delta^{k-1}[U_{j+1} - U_{j}]\text{for}k\geq 2.\]
III. RESULTS AND DISCUSSION
3.1 Preliminary Theorem
Suppose that and , for integral j.
Let be the set of even numbers and be the set of odd numbers.
Where fixed, , fixed.
Then,
\[\mathrm{(i)} _ {1} \Delta \bigl (\alpha_ {\mathrm{j}} \mathrm{U} _ {\mathrm{j}} \bigr) ^ {2} = \alpha_ {\mathrm{j}} ^ {2} (\mathrm{odd}) \text{for even case}\]
\[\mathrm{(i)} _ {2} \quad \Delta \bigl (\alpha_ {\mathrm{j}} \widehat{\mathrm{U}} _ {\mathrm{j}} + \beta \bigr) ^ {2} = \alpha_ {\mathrm{j}} ^ {2} (\mathrm{even}) \text{for odd case}\]
\[\Delta^ {2} \big (\alpha_ {\mathrm{j}} \widehat{\mathrm{U}} _ {\mathrm{j}} + \beta \big) ^ {2} = 8 = 2 ^ {2} (2!)\]
\[\Delta^ {\mathrm{k}} \big(\alpha_ {\mathrm{j}} \widehat{\mathrm{U}} _ {\mathrm{j}} + \beta_ {\mathrm{j}} \big) ^ {2} = 0, \mathrm{k} \in \{3, 4, \dots .\}\]
Proof:
Case 1. (even)
\[\text {Let} \langle \propto_ {\mathrm{j}} \mathrm{U} _ {\mathrm{j}} \rangle_ {\mathrm{j=0}} ^ {9} = \langle \propto_ {\mathrm{j}} \mathrm{j} \rangle_ {\mathrm{j=0}} ^ {9} = \{0, 2, 4, 6, 8, 1 0, 1 2, 1 4, 1 6, 1 8 \}\]
Then, the table below display the values of for selected values of and in the set and respectively when .
Table I: Difference Order Table for Selected Consecutive Positive Integers.
| $\alpha_j j$ | $(\alpha_j j)^2$ | $\Delta(\alpha_j j)^2$ | $\Delta^2(\alpha_j j)^2$ | $\Delta^3(\alpha_j j)^2$ |
| 0 | 0 | 4 | 8 | 0 |
| 2 | 4 | 12 | 8 | 0 |
| 4 | 16 | 20 | 8 | 0 |
| 6 | 36 | 28 | 8 | 0 |
| 8 | 64 | 36 | 8 | 0 |
| 10 | 100 | 44 | 8 | 0 |
| 12 | 144 | 52 | 8 | 0 |
| 14 | 196 | 60 | 8 | 0 |
| 16 | 256 | 68 | 8 | |
| 18 | 324 | | | |
Case II. (odd)
Let .
Then the table below displayed the values of for selected values of k and p in the set and respectively when .
Table II: Difference Order table for selected Consecutive Positive Integers
| $\alpha_{j} + \beta$ | $(\alpha_{j} + \beta)^{2}$ | $\Delta(\alpha_{j} + \beta)^{2}$ | $\Delta^{2}(\alpha_{j} + \beta)^{2}$ | $\Delta^{3}(\alpha_{j} + \beta)^{2}$ |
| 1 | 1 | 8 | 8 | 0 |
| 3 | 9 | 16 | 8 | 0 |
| 5 | 25 | 24 | 8 | 0 |
| 7 | 49 | 32 | 8 | 0 |
| 9 | 81 | 40 | 8 | 0 |
| 11 | 121 | 48 | 8 | 0 |
| 13 | 169 | 56 | 8 | 0 |
| 15 | 225 | 64 | 8 | |
| 17 | 289 | 72 | | |
| 19 | 361 | | | |
Similarly, it is clear that , for all , . The theorem is valid as observed from column 4 and 5.
The proof of (i), (ii) and (iii) are direct.
\[\begin{array}{r l} & {\mathrm{U_{j}} = \mathrm{j} \Rightarrow \alpha \mathrm{U_{j}} = \alpha_{\mathrm{j}} \Rightarrow (\alpha \mathrm{U_{j}})^{2} = \alpha^{2} \mathrm{j}^{2}} \\& {\Rightarrow (\alpha^{2} \mathrm{j}^{2}) = \alpha^{2} \Delta \mathrm{j}^{2} = \alpha^{2} [ (\mathrm{j} + 1)^{2} - \mathrm{j}^{2} ] = \alpha^{2} [ \mathrm{j}^{2} + 2 \mathrm{j} + 1 - \mathrm{j}^{2} ]} \\& {\qquad = \alpha^{2} (2 \mathrm{j} + 1) = \alpha^{2} (\mathrm{odd}) = \mathrm{even}.} \end{array}\tag{i1}\]
(i2)
\[\widehat {U} _ {j} = \widehat {j} \Rightarrow \propto \widehat {U} _ {j} + \beta = \propto \widehat {j} + \beta\]
\[\begin{array}{r l} & {\left(\widehat{U}_{j} + \beta\right)^{2} = \left(U_{\widehat{j}} + \beta\right)^{2} = \left(\alpha_{\widehat{j}}\right)^{2} + 2 \propto \beta_{j} + \beta^{2}} \\& {\Delta \left(\propto \widehat{U}_{j} + \beta_{j}\right)^{2} = \Delta \left(\alpha^{2} j^{2} + 2 \propto \beta_{j} + \beta^{2}\right)} \\& {\qquad = \alpha^{2} \Delta j^{2} + 2 \propto \beta \Delta_{j} + \beta^{2}} \\& {\qquad = \alpha^{2} [ (j + 1)^{2} - j^{2} ] + 2 \propto \beta [ (j + 1) - j ] + 0} \\& {\qquad = \alpha^{2} (j^{2} + 2 j + 1 - j^{2}) + 2 \propto \beta (1)} \\& {\qquad = \alpha^{2} (2 j + 1) + 2 \propto \beta} \\& {\qquad = 2 \alpha^{2} j + \alpha^{2} + 2 \propto \beta} \\& {\qquad = \alpha^{2} (2 j + 1) + 2 \propto \beta} \end{array}\]
\[\Rightarrow \Delta (\alpha \widehat{U}_{j} + \beta_{j})^{2} = \alpha^{2} (2 j + 1) + 2 \alpha \beta = even\]
For every , , .
\[\begin{array}{r l} \Delta^ {2} (\alpha U _ {j}) ^ {2} & = \Delta (\Delta (\alpha j)) ^ {2} = \Delta [ \alpha^ {2} (2 j + 1) ] \\& = \Delta (2 \alpha^ {2} j + \alpha^ {2}) = 2 \alpha^ {2} \Delta j + \Delta \alpha^ {2} \\& = 2 \alpha^ {2} [ (j + 1) - j ] + 0 \\& = 2 \alpha^ {2} (1) = 2 \alpha^ {2} \\\Rightarrow \Delta^ {2} (\alpha U _ {j}) ^ {2} & = 2 \alpha^ {2} \end{array}\tag{ii}\]
Similarly:
\[\Delta^ {2} \big (\propto \widehat{\mathrm{U}} _ {\mathrm{j}} + \beta \big) ^ {2} = \Delta [ \Delta (\propto \mathrm{j} + \beta) ^ {2} ] \\& = \Delta [ \propto^ {2} (2 \mathrm{j} + 1) + 2 \propto \beta ] \\& = \Delta [ 2 \propto^ {2} \mathrm{j} + \alpha^ {2} + 2 \propto \beta ]\]
\[= 2 \alpha^{2} \Delta j + \Delta\alpha^{2}+ 2\Delta\alpha\beta = 2 \alpha^{2} [(j + 1) - j] + 0 + 0 = 2 \alpha^{2}(1) = 2 \alpha^{2} \Rightarrow \Delta^{2}(\alpha \widehat{U}_{j} + \beta) = 2 \alpha^{2} \Rightarrow \Delta^{2}(\alpha U_{j})^{2} = \Delta^{2}(\alpha \widehat{U}_{j} + \beta) = 2 \alpha^{2}. \text{proving(ii)}\]
The principle of mathematical induction is needed in the proof of (iii).
\[For k = 3, \Delta^{3}(\alpha_{j} U_{j})^{2} = \Delta^{3}(\alpha j)^{2} = \Delta[\Delta^{2}(\alpha j)^{2}] = \Delta(2 \alpha^{2}) = 0\]
Similarly,
Consequently,
and
\[\Delta^{k}\left(\propto \widehat{U}_{j} + \beta\right) = \Delta^{k-3}\left[\Delta^{3}\left(\propto j + \beta\right)^{2}\right] = \Delta^{k-3}(0) = 0\]
proving , for all positive integer . This established the proof of (iii).
Thus, we have seen clearly that:
\[\text{Theorem 3.1} \Rightarrow \Delta(\propto U_{j})^{2} \text{and} \Delta(\propto \widehat{U}_{j} + \beta)^{2} \text{are even} \tag{i}\]
\[\Delta^{2}\big(\propto U_{j}\big)^{2} \text{and} \Delta^{2}\big(\propto \widehat{U}_{j} + \beta\big)^{2} = \alpha^{2}(2!), \alpha=2 \dots\dots\tag{ii}\]
\[\Delta^ {3} \left(\propto U _ {j}\right) ^ {2} \text {and} \Delta^ {3} \left(\propto \widehat {U} _ {j} + \beta\right) ^ {2} = 0 \not \sim k \geq 3 \dots\dots\tag{iii}\]
In the sequel, we examine the computational disposition of and
for every integral j and positive integral k and p. The results are summarized as in the following theorem.
3.2 Main Theorem
Let and , where is any integer. Then for arbitrary positive integer and , and is given by:
\[\Delta^{k} \big (\propto U_{j} \big)^{p} = \Delta^{k} \big (\propto \widehat{U}_{j} + \beta \big)^{p} \left\{
\begin{array}{l l}
0, & \text{if } k > p \quad (a) \\
\sum_{i = 1}^{p} {\binom{p}{i}} j^{i}, & \text{for } k = 1 \\
\propto^{k} k!, & \text{if } k = p \quad (c) \\
\text{even,} & \text{for } p \geq 1 \\
\text{even,} & \text{for } 2 \leq k < p
\end{array}
\right.
\tag{b}\]
(d) (e)
3.2.1 Proof of (a)
\[\begin{array}{l} \Delta^ {2} (\propto U _ {j}) = \Delta (\Delta \propto j) = \Delta [ \propto \Delta (j) ] = \Delta [ ((j + 1) - j) ] \\= \Delta [ \propto (1) ] = \Delta (\propto) = 0 \end{array}\]
Similarly,
\[\begin{array}{r l} \Delta^ {2} (\alpha \mathrm{U} _ {\mathrm{j}} + \beta) & = \Delta [ \Delta (\alpha \mathrm{j} + \beta) ] = \Delta [ \alpha \Delta \mathrm{j} + \Delta \beta ] \\& = \Delta [ \alpha ((\mathrm{j} + 1) - \mathrm{j}) ] = \Delta [ \alpha (1) ] \\& = \Delta (\alpha) = 0 \\\Delta^ {2} (\alpha \mathrm{U} _ {\mathrm{j}}) & = \Delta^ {2} (\alpha \mathrm{U} _ {\mathrm{j}} + \beta) = 0. \end{array}\]
So (a) is valid for k = 2 and p = 1, which are the least values for the respective exponents. Assume that (a) is valid for all pairs of integers , for which for some positive integers and such that , , p < k.
Then
and
(by induction hypothesis hypothesis) = 0. Therefore, the validity of (a) is established.
3.2.2 Proof of (b)
\[\begin{array}{l} (j + 1) ^ {p} = \sum_ {i = 1} ^ {p} \binom{p} {i} j ^ {i} = \sum_ {i = 1} ^ {p - 1} \binom{p} {i} j ^ {i} + \binom{p} {p} j ^ {p} \\\Rightarrow \quad (j + 1) ^ {p} - j ^ {p} = \sum_ {i = 1} ^ {p - 1} \binom{p} {i} j ^ {i} \quad \text{proving (b)} \end{array}\]
\[\Rightarrow \Delta^ {1} \bigl (\propto U _ {j} \bigr) ^ {p} = \propto^ {p} \left(\Delta U _ {j} ^ {p}\right) = \propto^ {p} \sum_ {i = 1} ^ {p - 1} {\binom{p} {i}} j ^ {i}\]
\[\text{and} \quad \Delta^{1} \bigl(\propto U_{j} + \beta\bigr)^{p} = \sum_{i=1}^{p-1} \binom{p}{i} (\propto j)^{p-i} = \propto^{p} \sum_{i=1}^{p-1} \binom{p}{i} j^{i}\]
Observe that, since , implies that:
\[\alpha \widehat{U}_{j} + \beta = \alpha j + 1\]
\[\therefore \quad (\alpha j + \beta)^{p} = \sum_{i=1}^{p-1} {\binom{p}{i}} (\alpha j)^{p-i} \beta^i = \sum_{i=1}^{p-1} {\binom{p}{i}} (\alpha j)^{p-i}\]
For simplicity of complexity of the odd form, we can logically express it as a single form with respect to even form, since they have same characteristic structures.
Claim that proof of even case is necessary and sufficient for the proof of the odd case.
Considering the even form for the remaining part of the proof proof even proof of odd.
We have that:
\[\Delta \big(\propto U_{j}\big)^{p} = \Delta \Big(\propto^{p} U_{j}^{p}\Big) = \propto^{p} \left(\Delta U_{j}^{p}\right) = \propto^{p} (\Delta j^{p}) \\= \propto^{p} [ (j + 1)^{p} - j^{p} ] = \propto^{p} \sum_{i=1}^{p-1} \binom{p}{i} j^{i}\]
3.2.3 Proof of (c)
We examine
\[\Rightarrow (\alpha U_{j}) = \alpha (\Delta j) = \alpha (j + 1 - j) = \alpha (1) = \alpha = 2!\]
\[\begin{array}{r l} \mathrm{k=2,} \Delta^ {2} \big (\alpha U_ {j} \big) ^ {2} & = \alpha^ {2} \Delta^ {2} U_ {j} ^ {2} = \Delta \alpha^ {2} (\Delta j^ {2}) = \alpha^ {2} \Delta [ (j+1) ^ {2} - j^ {2} ] \\& = \alpha^ {2} \Delta (j^ {2} + 2j+1-j^ {2}) = \alpha^ {2} \Delta (2j+1) \\& = \alpha^ {2} \Delta (2j+1) \\& = 2 \alpha^ {2} \Delta j + \Delta \alpha^ {2} \\& = 2 \alpha^ {2} (j+1-j) = 0 \\& = 2 \alpha^ {2} = \alpha^ {2} 2! = 2 ^ {2}. 2! \end{array}\]
\[\Rightarrow \Delta^{2} (\alpha U_{j})^{2} = \Delta^{2} (\alpha \widehat{U}_{j} + \beta) = \alpha^{2} 2! = 2^{2}.2! = 8.\]
by (ii) of theorem the theorem is valid for . Assume the validity of the theorem for for some integer . Then
by induction hypothesis. Finally, we need to prove that:
\[\Delta^ {q + 1} (\propto^ {q + 1} \mathrm{Uj} ^ {q + 1}) = \propto^ {q + 1} (q + 1)!\]
Claim is reproductive. Reproductive is understood to mean the following:
\[\Delta^{\mathrm{r}} \sum_{\mathrm{j} = 1}^{\mathrm{n}} \widehat{\alpha}_{\mathrm{j}} \mathbf{g}_{\mathrm{j}} (x) = \sum_{\mathrm{j} = 1}^{\mathrm{n}} \widehat{\alpha}_{\mathrm{j}} \Delta^{\mathrm{r}} \big(\mathbf{g}_{\mathrm{j}} (x)\big)\]
Where are arbitrary constants.
3.2.4 Proof of Claim
Consider .
\[\begin{array}{r l} & \mathrm{for~r = 1},\Delta(\widehat{\alpha}_1\mathrm{U}_{j_1}^{k_1} + \widehat{\alpha}_2\mathrm{U}_{j_2}^{k_2}) = \Delta(\widehat{\alpha}_1\mathrm{U}_{j_1}^{k_1} + \widehat{\alpha}_2\mathrm{U}_{j_2}^{k_2}) \\& \qquad = \widehat{\alpha}_1 (j + k)^{k_1} + \widehat{\alpha}_2 (j + k)^{k_2} - \widehat{\alpha}_1 \mathrm{U}_{j_1}^{k_1} - \widehat{\alpha}_2 \mathrm{U}_{j_2}^{k_2} \\& \qquad = \widehat{\alpha}_1 [(j_1 + 1)^{k_1} - j_i^{k_1}] + \widehat{\alpha}_2 [(j_2 + 1)^{k_2} - j_2^{k_2}] \\& \qquad = \widehat{\alpha}_1 \Delta(j_i^{k_1}) + \widehat{\alpha}_2 \Delta(j_2^{k_2}) \\& \qquad = \widehat{\alpha}_1 \Delta(\mathrm{U}_{j_i}^{k_1}) + \widehat{\alpha}_2 \Delta(\mathrm{U}_{j_2}^{k_2}) \end{array}\]
is a reproductive the claim is valid for , for some integer .
Then
Finally,
\[\begin{array}{r l} & {\Delta^ {t + 1} \big (\widehat {\alpha} _ {1} \mathrm{U} _ {j _ {1}} ^ {k _ {1}} + \widehat {\alpha} _ {2} \mathrm{U} _ {j _ {2}} ^ {k _ {2}} \big) = \Delta^ {t} \big [ \Delta \big (\widehat {\alpha} _ {1} \mathrm{U} _ {j _ {1}} ^ {k _ {1}} + \widehat {\alpha} _ {2} \mathrm{U} _ {j _ {2}} ^ {k _ {2}} \big) \big ]} \\& {= \Delta^ {t} \big (\widehat {\alpha} _ {1} \Delta \mathrm{U} _ {j _ {1}} ^ {k _ {1}} + \widehat {\alpha} _ {2} \Delta \mathrm{U} _ {j _ {2}} ^ {k _ {2}} \big) = \widehat {\alpha} _ {1} \Delta^ {t} \big (\Delta \mathrm{U} _ {j _ {1}} ^ {k _ {1}} \big) + \widehat {\alpha} _ {2} \Delta^ {t} \big (\Delta \mathrm{U} _ {j _ {2}} ^ {k _ {2}} \big)} \end{array}\]
(by induction hypothesis)
\[= \widehat{\alpha}_1 \Delta^{t+1} \big(\mathrm{U}_{j_1}^{k_1}\big) + \widehat{\alpha}_2 \Delta^{t+1} \big(\mathrm{U}_{j_2}^{k_2}\big)\]
So the theorem is valid for and hence valid for all positive integer r.
\[\begin{array}{r l} \mathrm{Now}, & \Delta^ {q + 1} (\propto^ {q + 1} \mathrm{Uj} ^ {q + 1}) = \Delta^ {q} [ \propto^ {q + 1} \Delta (\mathrm{Uj} ^ {q + 1}) ] \\& = \Delta^ {q} [ \propto^ {q + 1} ((\mathrm{j} + 1) ^ {q + 1} - \mathrm{j} ^ {q + 1}) ] \end{array}\]
\[= \Delta^{q} \left[ \propto^{q+1} \sum_{i = 1}^{q} \binom{q + 1}{i} j^{i} \right]\]
Therefore,
For every positive integer k, proving (c).
\[\begin{array}{r l r} & & {\Delta \big (\propto U _ {j} \big) ^ {p} = \propto^ {p} [ (j + 1) ^ {p} - j ]} \\& & {\Rightarrow \quad \propto^ {p} [ (j + 1) ^ {p} - j ] = \left\{ \begin{array}{l l} \mathrm{even-even=} \mathrm{evenforevencase} \\ \mathrm{odd-odd=oddforoddcase} \end{array} \right. } \\& & {\mathrm{Soinallscases} \Delta \big (\propto U _ {j} \big) ^ {p} \mathrm{and} \Delta \big (\propto \widehat{U} _ {j} + \beta \big) ^ {p} \mathrm{areeven,proving(d).}} \end{array}\]
3.2.6 Proof of (e)
Consider for . If , then the condition .
\[\Delta^{2} \big( \propto \mathrm{U}_{\mathrm{j}} \big)^{\mathrm{p}} = \Delta \big[ \Delta \big( \propto \mathrm{U}_{\mathrm{j}} \big)^{\mathrm{p}} \big]\]
(by the reproductive property.
\[= \alpha^{p} \left[ (j + 2)^{p} - (j + 1) - \left[ (j + 1)^{p} - j^{p} \right] \right]\]
So in all cases, and is even for .
Assume the validity of (e) for all positive integer k and p such that , for some positive integer m.
Then
\[\begin{array}{r l} \Delta^ {k + 1} \big (\propto U _ {j} \big) ^ {p} & = \Delta^ {k} \big (\Delta \big (\propto U _ {j} \big) ^ {p} \big) \\& = \Delta^ {k} \propto^ {p} \left[ [ (j + 2) ^ {p} - 2 (j + 1) ^ {p} + j ^ {p} ] \right] \\& = \Delta^ {k} \propto^ {p} \left[ [ (j + 2) ^ {p} ] - 2 \Delta^ {k} [ (j + 1) ^ {p} ] + \Delta^ {k} [ j ^ {p} ] \right] \\& \mathrm{(bythereproductivepropertyof} \Delta^ {k}) \\& \left(\Delta^ {k} \left(U _ {j + 2} ^ {p}\right) - 2 \Delta^ {k} \left(U _ {j + 1} ^ {p}\right) + \Delta^ {k} \left(U _ {j} ^ {p}\right)\right) \propto^ {p} \\& = \mathrm{even-even+even=even.} \\& = \propto^ {p} \Delta^ {k} \left[ U _ {j + 2} ^ {p} \right] - 2 \propto^ {p} \Delta^ {k} \left[ U _ {j + 1} ^ {p} \right] + \propto^ {p} \Delta^ {k} \left[ U _ {j} ^ {p} \right] \end{array}\]
Finally, we examine:
\[\Delta^{k} \big(\propto U_{j}\big)^{p+1}\]
\[\Delta^{k} \big(\propto U_{j}\big)^{p+1} = \Delta^{k} (\propto j)^{p+1} = \Delta^{k-1} [ \Delta (\propto j)^{p+1} ]\]
\[= \Delta^{k-1} [\alpha^{p+1} (j + 1)^{p+1} - 2(j + 1)^{p+1} + j^{p+1}]\]
\[= \Delta^{k-1} \left[ \alpha^{p+1} \left[ U_{j+2}^p \right] - 2 \alpha^{p+1} \Delta^{k-1} \left[ U_{j+1}^{p+1} \right] + \alpha^{p+1} \Delta^{k-1} \left[ U_j^{p+1} \right] \right]\]
\[= \text { even } - \text { even } + \text { even } = \text { even }.\]
(by the induction hypothesis). Since .
This completes the proof, that is the relation (e) hold for all +ve integers k and p for which , . Thus, the theorem is established.
3.3 Corollary
\[\Delta^ {k} (\propto j) ^ {k} = \Delta^ {k} \left(\propto^ {k} j ^ {k}\right) = \propto^ {k} \Delta^ {k} j ^ {k} = \propto^ {k} k!\]
For any integer j and for any positive integer k.
In other words, for any set of even and odd integers ( and ), the order difference of the power of any integers is equal to
The implication of (c) in theorem 3.2 is the following: "if any number of consecutive even or odd integers are raised to a positive integral power k, then the order difference is equal to
The existence of the triddle between the difference operator and the D operator (differential operator).
\[\mathrm{D}^{\mathrm{k}}(x^{\mathrm{p}}) = 0 \text{if} k > p\]
\[\mathrm{D}^{ ext{k}} \big( x^\text{k} \big) = \text{k}!\]
\[\mathrm{D}^\mathrm{k}\left(\propto^\mathrm{k} x^\mathrm{k}\right) = \propto^\mathrm{k} \mathrm{k}!\]
The coefficient of in :
\[D^{k}(x^{p}) = \frac{p!}{(p-k)!} x^{p-k} \text{for} k \geq 2, k < p \text{is even.}\]
\[\Rightarrow \alpha^{p} D^{k}(x^{p}) = \frac{\alpha^{p} p!}{(p-k)!} x^{p-k}, 2 \leq k < p \text{is even}\]
IV. CONCLUSION
This article established the structures of finite orders with respect to powers of consecutive elements of even and odd sets. Specifically, the results reveal a similarity between the difference orders and the D operator powers of monomials with positive integral powers as reflected in (a) and (c) of theorem 3.2 and (I) and (II) of remark 3.4.
