**3. Applications of the obtained results**

Direct use of formula (1) is performed on Appendix C in [2], alongside the use of another general formula for multinomial expansion, similar to (2), just obtained in the last section of this document. What is exposed here are applications not covered in [2]; these are regarded as theoretical and numerical applications and are given next.

### **3.1 Theoretical application**

<sup>¼</sup> <sup>X</sup>*<sup>δ</sup>* <sup>∗</sup> 0 *δ* ∗ <sup>1</sup> ¼0

> X*δ* ∗ 1 *δ* ∗ <sup>2</sup> ¼0

> > X *δ* ∗ 2

> > > X 1

*j*¼1

⋯ • X *δ* ∗ *s*�1

*δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ*<sup>s</sup>* <sup>þ</sup>1k*<sup>k</sup>*

*δ* ∗ *<sup>s</sup>* ¼0

X 1

*j*¼1

the function

*δ* ∗ <sup>3</sup> ¼0

*δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>3</sup> <sup>þ</sup>1k*<sup>k</sup>* X *δ* ∗ 0 1

8 >< >:

0 B@

> 0 B@

8 >< >: ⋯ X

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup>

⋯ X

*δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup>

*δ* ∗ 1

*δδ* <sup>∗</sup> <sup>1</sup> <sup>¼</sup><sup>1</sup>

> *δ* ∗ 2

*δδ* <sup>∗</sup> <sup>3</sup> <sup>¼</sup><sup>1</sup>

*j*,*k*,*::*, *δδ* <sup>∗</sup>

*δδ* <sup>∗</sup> <sup>2</sup> <sup>¼</sup><sup>1</sup>

⋯ X

*δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>3</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup>

<sup>1</sup> <sup>þ</sup><sup>1</sup>

*Mathematical Theorems - Boundary Value Problems and Approximations*

<sup>2</sup> <sup>þ</sup><sup>1</sup>

*δ* ∗ 3

<sup>3</sup> , … , *δδ* <sup>∗</sup> 2

⋯ X

*δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ*<sup>s</sup>* <sup>þ</sup>1k*δδ* <sup>∗</sup>

*j*,*k*,*::*, *δδ* <sup>∗</sup>

� � <sup>⋃</sup> f g*<sup>n</sup>* <sup>↦</sup> *<sup>h</sup>* <sup>⋃</sup>

X*δ* ∗ *f*�1 *δ* ∗ *<sup>f</sup>* ¼0

*j*,*k*,*::*, *δδ* <sup>∗</sup> *f* , … , *δδ* <sup>∗</sup> *f*�1

*ϕ*

*δϕ*¼1

þ1k*δϕ*þ<sup>1</sup>

<sup>¼</sup> <sup>Y</sup>*<sup>s</sup>*�<sup>1</sup> *f*¼1 ⋯ X 2

⋯ X 2

<sup>3</sup> <sup>þ</sup><sup>1</sup>

9 >=

>; *γ δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> 3 <sup>3</sup> *φ δ* ∗ 3

*δ* ∗ *s*

> 9 >=

>; *γ δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *s <sup>s</sup>*�<sup>1</sup> *<sup>φ</sup>*

*s j*¼1 *xj* � � <sup>⋃</sup> f g*<sup>n</sup>* !

> 8 >>><

0

BBB@

>>>:

X *δ* ∗ *f*�1 1

*δδ* <sup>∗</sup> *f*�1 ¼1

> 9 >>>=

1

CCCA*x δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f f*

>>>;

� � *k* ¼

*δδ* <sup>∗</sup> *<sup>s</sup>* <sup>¼</sup><sup>1</sup>

*<sup>s</sup>* , … , *δδ* <sup>∗</sup> *s*�1 *s* þ1

*δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*<sup>l</sup>*

*k*¼1

*k*¼1

⋯ X

*δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>3</sup> <sup>þ</sup>1k*<sup>l</sup>*

2

⋯ ••• ⋯••• ⋯ ••• ⋯••• ⋯ ••• ⋯

⋯ X

*δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ*<sup>s</sup>* <sup>þ</sup>1k*<sup>l</sup>*

2

⤸

<sup>¼</sup> <sup>X</sup> 1≤ *ϕ* ∈Z<sup>þ</sup> ≤ *s*

> þ1k*δδ* <sup>∗</sup> *<sup>f</sup>* <sup>þ</sup><sup>1</sup>

*δ* ∗ *f*

*δδ* <sup>∗</sup> *f* ¼1

3 7 7 7 5• *x δ* ∗ *<sup>s</sup>*�<sup>1</sup> *<sup>s</sup>* � �

X *ϕ*

*δδ* <sup>∗</sup> *f* ¼1 þ1

*<sup>ϕ</sup>* 6¼ *<sup>δ</sup>* <sup>∗</sup> *f*

*<sup>ϕ</sup>* <sup>¼</sup> *<sup>δ</sup>* <sup>∗</sup> *f*

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �

8

>>>>>>>>><

>>>>>>>>>:

X *ϕ δϕ*þ<sup>1</sup>

*δδ* <sup>∗</sup> *f* ¼1

( ) !*<sup>n</sup>*

��� <sup>X</sup>

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � � *x*f g*<sup>ϕ</sup>*

⋯ X

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �

2

*k*¼1

þ1k*l*

(26)

(27)

*k*¼1

*δ* ∗ *<sup>s</sup>*�<sup>1</sup> *<sup>s</sup>* � ⋯ � ⋯ � � ⋯ �

Then the expansion above follows a pattern that can be coded into a general formula; replace the variables *γ* and *φ* according to its definition; it would end with

f gn *π*<sup>þ</sup> ⊂*F X*½ � **A** :! *π*<sup>þ</sup> ⊂*F X*½ �

*k*¼1

X 1 X 0 1

*i*¼1 *ii* " #

X 0 1

*i*¼1 *ii* " # *j*,*k*,*::*, *δδ* <sup>∗</sup>

*j*, *k*,*::*, *δδ* <sup>∗</sup>

*<sup>s</sup>*�3• <sup>⋯</sup> (25)

<sup>1</sup> , … , *δδ* <sup>∗</sup> 0

<sup>2</sup> , … , *δδ* <sup>∗</sup> 1

9 >= >; *γ δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> 1 <sup>1</sup> •

9 >= >; *γ δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> 2 <sup>2</sup> •

*j*¼1

X 1

*j*¼1

⤸

*δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*<sup>k</sup>*

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>k</sup>*

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>l</sup>*

*δδ* <sup>∗</sup> <sup>0</sup> <sup>¼</sup><sup>1</sup>

X *δ* ∗ 1 1

*δδ* <sup>∗</sup> <sup>1</sup> <sup>¼</sup><sup>1</sup>

8 ><

0

B@

>:

X *δ* ∗ 2

*δδ* <sup>∗</sup> <sup>2</sup> <sup>¼</sup><sup>1</sup>

1

X 0 1

*i*¼1 *ii* " #

> 8 ><

0

B@

>:

X 0 1

*i*¼1 *ii* " #

⋃ *s j*¼1 *xj*

*x*f g*<sup>ϕ</sup>*

þ1k*k*

X 0 1

*i*¼1 *ii* " #

� � <sup>X</sup>

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �

X 1≤ *ϕ*∈ Z<sup>þ</sup> ≤*s*

!*<sup>n</sup>*

X 1

*j*¼1

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �

where

**104**

X *δ* ∗ *s*�1

*δδ* <sup>∗</sup> *s*�1 ¼1

1

A theoretical case application is exerted; the last results are performed to give an alternative proof for Lemma from [3].

**Lemma 3.1.** If *p* is a prime not dividing an integer *m*, then for all *n* ≥1, the binomial coefficient *<sup>p</sup>nm pn* � � is not divisible by *p.*

**Proof:** Formula (1) will be deployed for this purpose. Since *p* is prime, each summand in it arises from

$$\cdots \cdot \sum\_{\delta\_{\phi}=1}^{(p^{\star}m - p^{\star}) + 1 \parallel \delta\_{\phi+1}} \cdot \cdot i = \begin{cases} p^{\star(m-1)+1} \\ \sum\_{\delta\_{\phi}=1}^{\delta} & \phi = p^n \\ \sum\_{\delta\_{\phi+1}}^{\delta\_{\phi+1}} & \phi \neq p^n \end{cases} \tag{28}$$

(since *p* cannot be factored any further). Now, suppose for the sake of contradiction that there exists a prime *p* that divides the binomial coefficient the way it is proposed:

$$\left(\begin{array}{c} p^n m \\ p^n \end{array}\right) = \sum\_{\delta\_{p^n} = 1}^{p^n(m-1) + 1} \cdots \sum\_{1 \le j \le k \le \cdots \le \delta\_\delta \le (p^n m - p^\*) + 1} \left[ \sum\_{i=1}^1 i\_i \right] =: \Psi \left(p^n, m \right) = p \cdot x\_{\text{int}} \tag{29}$$

for some integer *xint* <sup>∈</sup> þ, where <sup>ψ</sup> *pn* ð Þ , *<sup>m</sup>* is defined to be the summation sequence function on the left side of (3).

$$\Rightarrow \sum\_{\delta\_{p^u}=1}^{p^u(m-1)+1} \cdots \sum\_{l=1}^{m} \sum\_{k=1}^{l} \sum\_{j=1}^{k} \left[\frac{1}{p}\right] = \mathcal{X}\_{int} \tag{30}$$

Since formula (1) represents addition of sequences of 1f g, expanding the above and gathering out the factors, the following would be obtained:

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$\left(\frac{1}{p}\right)(1) + \left(\frac{1}{p}\right)\left(p'' + \sum\_{i=0}^{p^n} (p'' - i) + \dots + k\_{p^n(m-1)+1}\right) = \infty\_{\text{int}}\tag{31}$$

But above there are no summands on the left side that yields an integer on the right side and at the same time fulfills:

$$\mathbf{1} + p^n + \sum\_{i=0}^{p^n} (p^n - i) + \dots + k\_{p^n(m-1)+1} = \mathbb{1}(p^n, m) \tag{32}$$

For if it were,

$$\Rightarrow \exists z \in \mathbb{Z}\_+ \models \psi(p^n, m) = z \; \forall m, n, p \in \mathbb{Z}\_+ : p \nmid m \tag{33}$$

so that, combining (4) and (5), it would be

$$\frac{1}{p} + \left(\frac{1}{p}\right)(z - 1) = \varkappa\_{\text{int}}$$

$$\Rightarrow \frac{\mathcal{Y}}{\not p} + \frac{z}{p} - \frac{\mathcal{Y}}{\not p} = \varkappa\_{\text{int}}\tag{34}$$

$$\Rightarrow \frac{z}{p} = \varkappa\_{\text{int}} \quad \Rightarrow \Leftarrow$$

which shows that indeed *xint* is all about a rational number. This contradicts the hypothesis, so the binomial coefficient is not divisible by *p*.

There is also problem 18.41 in [13] where the author introduces a case of study that takes place when *x* and *y* are members of a commutative ring of characteristic *p*:

**(Freshman exponentiation)**. Let p be prime. Show that in the ring *p*, we have

$$(a+b)^p = a^p + b^p \tag{35}$$

□

**107**

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation*

*DOI: http://dx.doi.org/10.5772/intechopen.91422*

for all *<sup>a</sup>*, *<sup>b</sup>*∈*<sup>p</sup>* [*hint:* observe that the usual binomial expansion for ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup> <sup>n</sup>* is valid in a *commutative* ring]. This one actually can alternatively be proven in a very similar way as the above lemma, so the proof is left to the reader. Some other study cases may come up that can be addressed with this result, where binomial coefficient calculation is part of their proof.

#### **3.2 Numerical application**

Two algorithms were written with the use of formulas (1) and (2); those were also implemented on two script programs written on the computer algebra system Maxima [14] and open-source software written in LISP [15] and based on a 1982 version of Macsyma [16]; of course there are many other CAS in which those can be implemented, i.e., here [17] is a great deal of one of them. The aim is to perform the expansion of a binomial by this result and perform the calculations of their individual coefficients. The first algorithm describes the calculation of binomial coefficients by formula (1), while the second is about the binomial and multinomial expansion based on formula (2); it also calculates the coefficients of the individual terms based on algorithm 1.

They are the exposed in the next two frames.

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation DOI: http://dx.doi.org/10.5772/intechopen.91422*

1 *p* � �

For if it were,

characteristic *p*:

cient calculation is part of their proof.

**3.2 Numerical application**

terms based on algorithm 1.

**106**

They are the exposed in the next two frames.

ð Þþ 1

right side and at the same time fulfills:

1 *p* � �

<sup>1</sup> <sup>þ</sup> *pn* <sup>þ</sup><sup>X</sup>

so that, combining (4) and (5), it would be

1 *p* þ

) �1 *p* þ *z p* � �1 *<sup>p</sup>* <sup>¼</sup> *xint*

) *z*

hypothesis, so the binomial coefficient is not divisible by *p*.

1 *p* � �

*<sup>p</sup>* <sup>¼</sup> *xint* )(

There is also problem 18.41 in [13] where the author introduces a case of study that takes place when *x* and *y* are members of a commutative ring of

which shows that indeed *xint* is all about a rational number. This contradicts the

**(Freshman exponentiation)**. Let p be prime. Show that in the ring *p*, we have

for all *<sup>a</sup>*, *<sup>b</sup>*∈*<sup>p</sup>* [*hint:* observe that the usual binomial expansion for ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup> <sup>n</sup>* is valid in a *commutative* ring]. This one actually can alternatively be proven in a very similar way as the above lemma, so the proof is left to the reader. Some other study cases may come up that can be addressed with this result, where binomial coeffi-

Two algorithms were written with the use of formulas (1) and (2); those were also implemented on two script programs written on the computer algebra system Maxima [14] and open-source software written in LISP [15] and based on a 1982 version of Macsyma [16]; of course there are many other CAS in which those can be implemented, i.e., here [17] is a great deal of one of them. The aim is to perform the expansion of a binomial by this result and perform the calculations of their individual coefficients. The first algorithm describes the calculation of binomial coefficients by formula (1), while the second is about the binomial and multinomial expansion based on formula (2); it also calculates the coefficients of the individual

ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup> <sup>p</sup>* <sup>¼</sup> *ap* <sup>þ</sup> *<sup>b</sup><sup>p</sup>* (35)

*pn*

*i*¼0

*pn* <sup>þ</sup><sup>X</sup> *pn*

*Mathematical Theorems - Boundary Value Problems and Approximations*

*i*¼0

*pn* ð Þþ � *<sup>i</sup>* <sup>⋯</sup> <sup>þ</sup> *kpn*ð Þþ *<sup>m</sup>*�<sup>1</sup> <sup>1</sup>

*<sup>p</sup><sup>n</sup>* ð Þþ � *<sup>i</sup>* <sup>⋯</sup> <sup>þ</sup> *kpn*ð Þþ *<sup>m</sup>*�<sup>1</sup> <sup>1</sup> <sup>¼</sup> <sup>ψ</sup> *pn* ð Þ , *<sup>m</sup>* (32)

) <sup>∃</sup>*z*∈þ∍<sup>ψ</sup> *pn* ð Þ¼ , *<sup>m</sup> <sup>z</sup>* <sup>∀</sup>*m*, *<sup>n</sup>*, *<sup>p</sup>* <sup>∈</sup><sup>þ</sup> : *<sup>p</sup>* <sup>∤</sup> *<sup>m</sup>* (33)

ð Þ¼ *z* � 1 *xint*

¼ *xint* (31)

(34)

□

!

But above there are no summands on the left side that yields an integer on the
