**2.2 Extension to** *n*�**summands**

By the Lemma 2.3 we can construct

*f* : <sup>þ</sup> ! ⋃

by *f x*ð Þ ≔ ⋃

Then the composition *<sup>f</sup>* ◦ *<sup>g</sup>* will be the searched function:

*Mathematical Theorems - Boundary Value Problems and Approximations*

by ð Þ *<sup>f</sup>* ◦ *<sup>g</sup>* ð Þ¼ *<sup>x</sup>* <sup>⋃</sup>

This could be graphically represented in **Figure 1**.

countably infinite if there is a bijective correspondence:

In the next theorem, the collection *δϕ* is countable:

Then S is countable [9].

**Figure 1.**

**102**

that the summands in (1) are also countable.

*Graphical representation of the mapping f* ð Þ ◦ *<sup>g</sup>* : <sup>þ</sup> ! *δϕ*.

*j* ∈*J* f g*z <sup>j</sup>*

f g*z <sup>j</sup>*

f g*<sup>z</sup> <sup>j</sup>* <sup>⋃</sup> f g *Im g*ð Þ (16)

*f* : *A* ! <sup>þ</sup> (17)

*En* (18)

(15)

*j* ∈L ⌊ *<sup>x</sup>* <sup>19</sup> ð Þ⌋

ð Þ *<sup>f</sup>* ◦ *<sup>g</sup>* : <sup>þ</sup> ! *δϕ*

*j* ∈ L ⌊ *<sup>x</sup>* <sup>19</sup> ð Þ⌋

**Definition 2.4.** A set *A* is said to be infinite if it is not finite [8]. It is said to be

**Theorem 2.5.** Let f g *En* , *n* ¼ 1, 2, 3, …, be a sequence of countable sets, and put

Since it was possible to establish a bijection between <sup>þ</sup> and the collection *δϕ*, and since each summation in (1) corresponds to just one element of *δϕ*, it follows

*S* ¼ ⋃ ∞ *n*¼1

Now that the countability of the collection *δϕ* was explained and a bijective function settled down to perform it, we will proceed to extend (1) to *n*� summands. Note that in [2] this formula remained unaltered in this sense, so that it was not extended; that is why we will do it here.

For the foremost part, another index set on a half-open interval is introduced, defined in similar way as *δϕ*:

$$\delta\_{\phi}^{\*} \coloneqq \{ [\hat{n}, i^{\*}, j^{\*}, k^{\*}, \dots z^{\*}, zh^{\*}, zi^{\*}, \dots, +\infty) : 0 \le \phi < +\infty; \phi \in \mathbb{Z}\_{+} \} \subset \mathbb{Z}\_{+} \tag{19}$$

The need of another index set comes from the fact that in the extension to the *n*� summands approach, a second lawyer of summands sequences and a new sequence of multipliers will arise; this way index scrambling between the two lawyers will be avoided. Then in a similar fashion, the modified function will apply:

$$(f' \circ \mathbf{g'}) : \mathbb{Z}\_+ \to \delta^\*\_{\phi} \tag{20}$$

where *f* <sup>0</sup> and *g*<sup>0</sup> are defined the same way as in the former, regarding ∗ as a superscript on the alphabetic letters.

**Theorem 2.7. Principle of recursive definition**. Let *A* be a set; let *a*<sup>0</sup> be an element of *A*. Suppose *ρ* is a function that assigns to each function *f* mapping a nonempty section of positive integers into *A*, an element of *A*. Then there exists a unique function

$$h: \mathbb{Z}\_+ \to \mathbf{A} \tag{21}$$

such that *h*ð Þ¼ 1 *a*0,

$$h(i) = \rho(h \mid \{1, \ldots, i - 1\}) \; \forall i > 1. \tag{22}$$

Now the same method developed in [2] will be performed, following the binomial theorem [10–12] and the recursive principle: Let *φ*, *γ* ∈P*n*ð Þn *F* **A** be two collection of summands; set the following:

$$\boldsymbol{\rho}\_{\{\boldsymbol{s}-\boldsymbol{f}\}} = \sum\_{\boldsymbol{\delta}\_{\boldsymbol{f}}^{\*} - \boldsymbol{0}\big|\_{\boldsymbol{\lambda}=\boldsymbol{0}} \leq \boldsymbol{\phi}\_{\boldsymbol{s}} \leq \boldsymbol{\delta}\_{\boldsymbol{f}-1}^{\*}\big|\_{\boldsymbol{\lambda}=\boldsymbol{0}}} \binom{\boldsymbol{\phi}\big|\_{\boldsymbol{\lambda}=\boldsymbol{1}}}{\boldsymbol{\phi}\big|\_{\boldsymbol{\lambda}=\boldsymbol{0}}} \boldsymbol{\eta}\_{\boldsymbol{f}}^{\boldsymbol{\phi}\big|\_{\boldsymbol{\lambda}=\boldsymbol{1}} - \boldsymbol{\phi}\big|\_{\boldsymbol{s}=\boldsymbol{0}}} \boldsymbol{\rho}\_{\{\boldsymbol{s}-\boldsymbol{f}\}}^{\boldsymbol{\phi}\big|\_{\boldsymbol{s}=\boldsymbol{0}}}\tag{23}$$

Subindices f g*f* , f g *s* � *f* ∈<sup>þ</sup> represent a consecutive number of a summand and the amount of remaining summands after the binomial theorem expansion, respectively. Continue recursively performing the expansion:

$$\sum\_{\delta\_1^\*=0}^{\delta\_0^\*} \left\{ \sum\_{\delta\_{\delta\_0^\*}^\*=1}^1 \dots \sum\_{\delta\_{\delta\_1^\*}^\*=1}^{\{\delta\_0^\* - \delta\_1^\*\} + 1 \|\delta\_{\delta\_1^\*} + 1\|} \dots \sum\_{k=1}^{\{\delta\_0^\* - \delta\_1^\*\} + 1 \|l} \sum\_{j=1}^{\{\delta\_0^\* - \delta\_1^\*\} + 1 \|k} \left[ \sum\_{i=1}^1 i\_0 \right]\_{j,k,\dots,\delta\_{\delta\_1^\*},\dots,\delta\_{\delta\_0^\*}} \right\} \gamma\_1^{\delta\_0^\* - \delta\_1^\*} \rho\_{s-1}^{\delta\_1^\*} \tag{24}$$

<sup>¼</sup> <sup>X</sup>*<sup>δ</sup>* <sup>∗</sup> 0 *δ* ∗ <sup>1</sup> ¼0 X *δ* ∗ 0 1 *δδ* <sup>∗</sup> <sup>0</sup> <sup>¼</sup><sup>1</sup> ⋯ X *δ* ∗ 1 *δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup> <sup>1</sup> <sup>þ</sup><sup>1</sup> *δδ* <sup>∗</sup> <sup>1</sup> <sup>¼</sup><sup>1</sup> ⋯ X 2 *δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>l</sup> k*¼1 X 1 *δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>k</sup> j*¼1 X 0 1 *i*¼1 *ii* " # *j*,*k*,*::*, *δδ* <sup>∗</sup> <sup>1</sup> , … , *δδ* <sup>∗</sup> 0 8 >< >: 9 >= >; *γ δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> 1 <sup>1</sup> • 0 B@ X*δ* ∗ 1 *δ* ∗ <sup>2</sup> ¼0 X *δ* ∗ 1 1 *δδ* <sup>∗</sup> <sup>1</sup> <sup>¼</sup><sup>1</sup> ⋯ X *δ* ∗ 2 *δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup> <sup>2</sup> <sup>þ</sup><sup>1</sup> *δδ* <sup>∗</sup> <sup>2</sup> <sup>¼</sup><sup>1</sup> ⋯ X 2 *δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*<sup>l</sup> k*¼1 X 1 *δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*<sup>k</sup> j*¼1 X 0 1 *i*¼1 *ii* " # *j*, *k*,*::*, *δδ* <sup>∗</sup> <sup>2</sup> , … , *δδ* <sup>∗</sup> 1 0 B@ 9 >= >; *γ δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> 2 <sup>2</sup> • 8 >< >: X *δ* ∗ 2 *δ* ∗ <sup>3</sup> ¼0 X *δ* ∗ 2 1 *δδ* <sup>∗</sup> <sup>2</sup> <sup>¼</sup><sup>1</sup> ⋯ X *δ* ∗ 3 *δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>3</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup> <sup>3</sup> <sup>þ</sup><sup>1</sup> *δδ* <sup>∗</sup> <sup>3</sup> <sup>¼</sup><sup>1</sup> ⋯ X 2 *δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>3</sup> <sup>þ</sup>1k*<sup>l</sup> k*¼1 ⤸ 8 >< >: 0 B@ X 1 *δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>3</sup> <sup>þ</sup>1k*<sup>k</sup> j*¼1 X 0 1 *i*¼1 *ii* " # *j*,*k*,*::*, *δδ* <sup>∗</sup> <sup>3</sup> , … , *δδ* <sup>∗</sup> 2 9 >= >; *γ δ* ∗ <sup>2</sup> �*<sup>δ</sup>* <sup>∗</sup> 3 <sup>3</sup> *φ δ* ∗ 3 *<sup>s</sup>*�3• <sup>⋯</sup> (25) ⋯ ••• ⋯••• ⋯ ••• ⋯••• ⋯ ••• ⋯ ⋯ • X *δ* ∗ *s*�1 *δ* ∗ *<sup>s</sup>* ¼0 X *δ* ∗ *s*�1 1 *δδ* <sup>∗</sup> *s*�1 ¼1 ⋯ X *δ* ∗ *s δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ*<sup>s</sup>* <sup>þ</sup>1k*δδ* <sup>∗</sup> *s* þ1 *δδ* <sup>∗</sup> *<sup>s</sup>* <sup>¼</sup><sup>1</sup> ⋯ X 2 *δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ*<sup>s</sup>* <sup>þ</sup>1k*<sup>l</sup> k*¼1 ⤸ 8 >< >: 0 B@ X 1 *δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ*<sup>s</sup>* <sup>þ</sup>1k*<sup>k</sup> j*¼1 X 0 1 *i*¼1 *ii* " # *j*,*k*,*::*, *δδ* <sup>∗</sup> *<sup>s</sup>* , … , *δδ* <sup>∗</sup> *s*�1 9 >= >; *γ δ* ∗ *<sup>s</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *s <sup>s</sup>*�<sup>1</sup> *<sup>φ</sup> δ* ∗ *<sup>s</sup>*�<sup>1</sup> *<sup>s</sup>* � ⋯ � ⋯ � � ⋯ �

This general formula actually performs the multinomial expansion along with

Since (2) represents finite products of countable sets (as it was exposed previously), it follows from theorem 2.8 that the sequence of multipliers in (2) is also

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

A theoretical case application is exerted; the last results are performed to give an

**Lemma 3.1.** If *p* is a prime not dividing an integer *m*, then for all *n* ≥1, the

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

8 >>>>>><

>>>>>>:

(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

for some integer *xint* <sup>∈</sup> þ, where <sup>ψ</sup> *pn* ð Þ , *<sup>m</sup>* is defined to be the summation

X 2 *l*

*k*¼1

Since formula (1) represents addition of sequences of 1f g, expanding the above

⋯X 3 *m*

*l*¼1

X *ϕ pn*ð Þþ *<sup>m</sup>*�<sup>1</sup> <sup>1</sup>

*δϕ*¼1

X *ϕ δϕ*þ<sup>1</sup>

*δϕ*¼1

X 0 1

*i*¼1 *ii* " #

> X 1 *k*

1 *p* � �

*j*¼1

*<sup>ϕ</sup>* <sup>¼</sup> *pn*

<sup>≕</sup> <sup>ψ</sup> *<sup>p</sup><sup>n</sup>* ð Þ¼ *; <sup>m</sup> <sup>p</sup>* � *xint* (29)

¼ *xint* (30)

(28)

*<sup>ϕ</sup>* 6¼ *pn*

� � *i* ¼

� � is not divisible by *p.*

the calculation of the coefficients of individual terms, for an expression of

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation*

**Theorem 2.8.** A finite product of countable sets is countable [5].

*n*�summands; its proof is given on Appendix A.

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

**3. Applications of the obtained results**

**3.1 Theoretical application**

binomial coefficient *<sup>p</sup>nm*

summand in it arises from

proposed:

**105**

*pnm pn* � � <sup>¼</sup> <sup>X</sup> *pn <sup>p</sup>n*ð Þþ *<sup>m</sup>*�<sup>1</sup> <sup>1</sup>

*δpn*¼1

sequence function on the left side of (3).

alternative proof for Lemma from [3].

*pn*

� � <sup>X</sup>

*ϕ <sup>p</sup>nm*�*pn* ð Þþ1k*δϕ*þ<sup>1</sup>

*δϕ*¼1

⋯ X

) <sup>X</sup>

*pn <sup>p</sup>n*ð Þþ *<sup>m</sup>*�<sup>1</sup> <sup>1</sup>

and gathering out the factors, the following would be obtained:

*δpn*¼1

<sup>1</sup><sup>≤</sup> *<sup>j</sup>*<sup>≤</sup> *<sup>k</sup>*≤⋯≤*δϕ* <sup>≤</sup> *pnm* � *pn* ð Þþ <sup>1</sup> *δpnm* ¼ 1∀*δϕ*

countable.

next.

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 the function

$$\{\pi\_{+} \in F[X]\} \bigvee \omega^{\prime} := \pi\_{+} \subset F[X]$$

$$\bigcup\_{j=1}^{i} \{\boldsymbol{\mathfrak{x}}\_{j}^{\cdot}\} \bigcup \{n\} \mapsto \left\{ h \left( \bigcup\_{j=1}^{i} \{\boldsymbol{\mathfrak{x}}\_{j}^{\cdot}\} \bigcup \{n\} \right) = \left( \sum\_{1 \le \boldsymbol{\phi} \in \mathbb{Z}\_{+} \ \boldsymbol{\le} \boldsymbol{\phi}} \boldsymbol{\mathfrak{x}}\_{\{\boldsymbol{\phi}\}} \right)^{n} \right\}$$

$$\left( \sum\_{1 \le \boldsymbol{\phi} \in \mathbb{Z}\_{+} \ \boldsymbol{\le} \boldsymbol{\le}} \boldsymbol{\mathfrak{x}}\_{\{\boldsymbol{\phi}\}} \right)^{n} = \prod\_{f=1}^{i-1} \left[ \sum\_{\boldsymbol{\delta}\_{f}^{\prime} = 0}^{\delta\_{f-1}^{\prime}} \left( \left\{ \sum\_{\begin{subarray}{c} \delta\_{f-1}^{\prime} \\ \delta\_{\delta\_{f-1}^{\prime}}^{\prime} \end{subarray}} \dots \sum\_{\delta\_{f}^{\prime} = 1}^{\delta\_{f}^{\prime}} \dots \sum\_{\delta\_{f}^{\prime}} \right. \right. \\ \left. \left( \sum\_{\begin{subarray}{c} \delta\_{f}^{\prime} = 1 \\ \delta\_{f}^{\prime} = 1 \end{subarray}} \boldsymbol{\delta\_{f}} \right) \right) \mathbbm{1}\_{j,k}^{\delta\_{f}^{\prime} } \dots \left( \boldsymbol{\$$

where

$$\cdots \quad \sum\_{\delta\_{\phi}=1}^{\left(\delta\_{f-1}^{\*}-\delta\_{f}^{\*}\right)+1\|\delta\_{\phi+1}} \quad \cdot k = \begin{cases} \left(\delta\_{f-1}^{\*}-\delta\_{f}^{\*}\right)+1 \\ \sum\_{\delta\_{\phi}=1}^{\phi} & \Phi = \delta\_{f}^{\*} \\ \sum\_{\delta\_{\phi}=1}^{\delta\_{\phi+1}} & \Phi \neq \delta\_{f}^{\*} \end{cases} \tag{27}$$

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

This general formula actually performs the multinomial expansion along with the calculation of the coefficients of individual terms, for an expression of *n*�summands; its proof is given on Appendix A.

**Theorem 2.8.** A finite product of countable sets is countable [5].

Since (2) represents finite products of countable sets (as it was exposed previously), it follows from theorem 2.8 that the sequence of multipliers in (2) is also countable.
