**Appendix A: a proof for formula (2)**

Here a proof for formula (2) is provided; let us proceed by induction on *s*. For *s* amount of summands, its expansion by (2) would be given by the following equation: *Alternative Representation for Binomials and Multinomies and Coefficient Calculation DOI: http://dx.doi.org/10.5772/intechopen.91422*

$$(\mathbf{x}\_1 + \mathbf{x}\_2 + \dots + \mathbf{x}\_i)^n = \prod\_{f=1}^{i-1} \left[ \sum\_{\delta\_f^+=0}^{\delta\_f^+} \left( \left\{ \sum\_{\delta\_{f-1}^+=1}^1 \dots \sum\_{\delta\_f^+=-1}^{\left(\delta\_{f-1}^+ - \delta\_f^+\right) + 1 \mid \delta\_{f-1}^+} \dots \sum\_{\delta\_{f-1}^+=1}^{\left(\delta\_{f-1}^+ - \delta\_f^+\right) + 1 \mid \delta\_f^+} \right\} \right)$$

$$\left( \sum\_{j=1}^{\delta\_{f-1}^+ - \delta\_f^+} \left[ \sum\_{i=1}^1 i\_i \right]\_{j, k, \dots, \delta\_f^+, \dots, \delta\_{f-1}^+} \right) \right) \mathbf{x}\_f^{\delta\_{f-1}^+ - \delta\_f^+} \left[ \bullet \left( \mathbf{x}\_i^{\delta\_{f-1}^+} \right) \right]$$

Fix s = 2 on (6); then the following is obtained:

$$\begin{split} \left(\mathbf{x}\_{1} + \mathbf{x}\_{2}\right)^{n} &= \prod\_{j=1}^{1} \left[ \sum\_{\delta\_{j}^{\*}=0}^{\delta\_{j-1}^{\*}} \left( \left\{ \sum\_{\begin{subarray}{c} \delta\_{j-1}^{\*} \\ \delta\_{j-1}^{\*} \end{subarray}}^{1} \dots \sum\_{\begin{subarray}{c} \delta\_{j}^{\*} \\ \delta\_{j}^{\*} \end{subarray}}^{1} \dots \sum\_{\begin{subarray}{c} \delta\_{j}^{\*} \\ \delta\_{j}^{\*} \end{subarray}}^{1} \dots \sum\_{\begin{subarray}{c} \delta\_{j}^{\*} \\ \delta\_{j}^{\*} \end{subarray}}^{1} \dots \sum\_{\begin{subarray}{c} \delta\_{j}^{\*} \\ \delta\_{j}^{\*} \end{subarray}}^{1} \dots \sum\_{\begin{subarray}{c} \delta\_{j}^{\*} \\ \delta\_{j}^{\*} \end{subarray}}^{1} \right) \right] \end{split} \tag{37}$$
 
$$\left(\sum\_{j=1}^{\delta\_{j-1}^{\*} - \delta\_{j}^{\*}} \right)^{+1} \mathbf{1}\_{j}^{\left[1 \right]} \left[ \sum\_{i=1}^{1} \mathbf{o}\_{i} \right]\_{j,k,\ldots,\delta\_{j}^{\*},\ldots,\delta\_{j-1}^{\*}} \right) \Bigg) \mathbf{x}\_{j}^{\delta\_{j-1}^{\*} - \delta\_{j}^{\*}} \Bigg] \mathbf{x}\_{j}^{\delta\_{j-1}^{\*} - \delta\_{j}^{\*}} \Bigg[ \mathbf{x} \left( \mathbf{x}\_{1}^{\delta\_{1}^{\*} - 1} \right) $$
 
$$= \sum\_{i=0}^{n} \binom{n}{i} \mathbf{x}\_{1}^{n-i} \mathbf{x}\_{2}^{i}$$

that actually stands for the binomial expansion for 2�summands; then it is correct. Now, fix *s* ¼ *k* on (6) and assume by hypothesis that it is correct.

$$\left(\left(\mathbf{x}\_1 + \mathbf{x}\_2 + \dots + \mathbf{x}\_k\right)^n\right)^n = \prod\_{f=1}^{k-1} \left[\sum\_{\delta\_f^\*=0}^{\delta\_{f-1}^\*} \left(\binom{\delta\_{f-1}^\*}{\delta\_f^\*}\right) \mathbf{x}\_f^{\delta\_{f-1}^\* - \delta\_f^\*}\right] \bullet \left(\mathbf{x}\_k^{\delta\_{k-1}^\*}\right) \tag{38}$$

If formula (2) is correct, it must be likewise valid for *s* ¼ *k* þ 1. Following up, the attempt is to prove that

$$\prod\_{f=1}^{(k+1)-1} \left[ \sum\_{\delta\_f^\*=0}^{\delta\_{f-1}^\*} \left( \binom{\delta\_{f-1}^\*}{\delta\_f^\*} \right) x\_f^{\delta\_{f-1}^\*-\delta\_f^\*} \right] \bullet \left( x\_{k+1}^{\delta\_{(k+1)-1}^\*} \right) = \left( \varkappa\_1 + \varkappa\_2 + \dots + \varkappa\_k + \varkappa\_{k+1} \right)^n \tag{39}$$

First, it will be expanded to the left side of the above equality:

$$\sum\_{\delta\_1=0}^{\delta\_0^\*} \left\{ \left\{ \sum\_{\delta\_0^\*=1}^1 \dots \sum\_{\delta\_{\delta\_1^\*}^{\delta\_1^\*}} \dots \sum\_{\delta\_{\delta\_1^\*}^{\delta\_1^\*}} \dots \sum\_{k\_1^\*}^{\delta\_1^\*} \dots \sum\_{k\_1^\*}^{\delta\_1^\*} \sum\_{j=1}^{\|\mathbf{l}\|} \left[ \sum\_{i=1}^1 a\_i \right]\_{j,k,\ldots,\delta\_{\delta\_1^\*}^{\delta\_1^\*},\dots,\delta\_{\delta\_\delta^\*}^\*} \right\} \right\} Y\_1^{\delta\_1^\* \cdots \delta\_1^\*} \bullet$$

$$\sum\_{\delta\_2^\*=0}^{\delta\_1^\*} \left\{ \left\{ \sum\_{\delta\_1^\*=1}^1 \dots \sum\_{\delta\_{\delta\_1^\*}^{\delta\_1^\*}} \dots \sum\_{\delta\_{\delta\_2^\*}^{\delta\_2^\*}} \dots \sum\_{k\_1^\*}^{\delta\_2^\*} \sum\_{j=1}^{\|\mathbf{l}\|} \left[ \sum\_{i=1}^{\delta\_1^\*-\delta\_2^\*} \delta\_i \right]\_{j,k,\ldots,\delta\_{\delta\_1^\*}^\*,\dots,\delta\_{\delta\_1^\*}^\*} \right\} Y\_2^{\delta\_1^\* \cdots \delta\_1^\*} \bullet \dots \bullet$$

$$\dots \bullet \dots \bullet \dots \bullet \dots \bullet \dots \bullet \dots \cdots$$

$$\dots \bullet \sum\_{\delta\_k^\*=0}^{\delta\_{k-1}^\*} \left( \left\{ \sum\_{\delta\_{k-1}^\*=1}^1 \dots \sum\_{\delta\_k^\*$$

the developing and programming of an algorithm to implement the use of the general formula obtained in [3], whose output would be something very similar to the one presented here. But overall, some algorithm could be raised, targeted at speed of calculations, to see if this method can be at least as fast as the current ones

*Mathematical Theorems - Boundary Value Problems and Approximations*

I want to thank my boss, Engineer Eloy Cavazos Galindo, for his constant support and advice during the time I've been working in Trefilados Plant, and for giving me the opportunity to further prepare myself by studying for my master's

<sup>0</sup> Union of the set of natural numbers with the zero element

•,•⋯• Denote a product of a summation sequence and a sequence of those products, respectively

*a*k*b* Logic operator OR; it is equivalent to *a*∨*b* (used this way

*<sup>i</sup>*¼1*<sup>c</sup> ii*,*j*,*k*, … Sum of the first *<sup>a</sup>* or *<sup>b</sup>* numbers. The subindexes *<sup>i</sup>*, *<sup>j</sup>*, *<sup>k</sup>*, …

⤸ Indicates a sequence of summations continues on the

Here a proof for formula (2) is provided; let us proceed by induction on *s*. For *s* amount of summands, its expansion by (2) would be given by the following equation:

*<sup>ϕ</sup>* , … Same as above but within (\*) superscript to contradistin-

indicate the different indexes of summations

corresponding to the sequence it belongs to, where *c* ¼ 0, 1, 2, … denotes the actual number of a summation

Sum of the *ϕ*-th *x* plus the *ϕ*-th *y* elements, raised to the

Sum of one to one; it equals the unity, written this way to set a starting point and give logic continuity to a sum

*k* Entries of binomial or multinomial coefficients

⋯ Denotes that more summations follow a sequence

guish from the above

due to space reasons) ⋯PPP Summation sequence (two or more nested operators)

sequence

*a*-power

sequence *∂* The order of a polynomial

next row

� � Binomial coefficient

L The generated of a set

**Appendix A: a proof for formula (2)**

*δϕ* Represents the *ϕ*-th alphabetic character

*i*, *j*, *k*, … , *δϕ*, … Indexes of summations

or even faster.

**Thanks**

degree.

*n*^, ^

*i* <sup>∗</sup> , *j*

P*<sup>a</sup>*k*<sup>b</sup>*

*a b*

P<sup>1</sup> *<sup>i</sup>*¼<sup>1</sup>*ii* h i

**110**

*<sup>x</sup>*f g*<sup>b</sup>* <sup>þ</sup> *<sup>y</sup>*f g*<sup>b</sup>* � �*<sup>a</sup>*�

� � *b*¼*ϕ*

*j*,*k*, …

<sup>∗</sup> , *k* <sup>∗</sup> , … , *δ* <sup>∗</sup>

**Notations**

X 1 *δ* ∗ *<sup>k</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ *<sup>k</sup>* <sup>þ</sup>1k*<sup>k</sup> j*¼1 X 0 1 *i*¼1 *ii* " # *j*,*k*,*::*, *δδ* <sup>∗</sup> *k* , … , *δδ* <sup>∗</sup> *k*�1 9 >= >; *γ δ* ∗ *<sup>k</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *k <sup>k</sup>*�<sup>1</sup> *<sup>φ</sup> δ* ∗ *k*�1 *k* � � �� ⋯ � � ⋯ � ) <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> <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 8 >< >: 9 >= >; *γ δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> 2 <sup>2</sup> •⋯ 0 B@ ⋯ ••• ⋯••• ⋯ ••• ⋯••• ⋯ ⋯ • *δ* X ∗ ð Þ� *k*þ1 2 *δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>1</sup>¼<sup>0</sup> X *δ* ∗ ð Þ� *k*þ1 2 1 *δδ* <sup>∗</sup> ð Þ� *k*þ1 2 ¼1 ⋯ X *δ* ∗ ð Þ� *k*þ1 1 *δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup> ð Þ� *k*þ1 1 � �þ1k*δδ* <sup>∗</sup> ð Þ� *k*þ1 1 þ1 *δδ* <sup>∗</sup> ð Þ� *k*þ1 1 ¼1 ⋯ X 2 *δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup> ð Þ� *k*þ1 1 � �þ1k*l <sup>k</sup>*¼<sup>1</sup> <sup>⤸</sup> 8 >>>< >>>: 0 BBB@ X 1 *δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup> ð Þ� *k*þ1 1 � �þ1k*k j*¼1 X 0 1 *i*¼1 *ii* " #*j*, *k*,*::*, *δδ* <sup>∗</sup> ð Þ� *k*þ1 1 , … , *δδ* <sup>∗</sup> ð Þ� *k*þ1 2 9 >= >; *γ δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup> ð Þ� *k*þ1 1 ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup> *<sup>φ</sup> δ* ∗ ð Þ� *k*þ1 2 ð Þ� *k*þ1 1 � � � 1 CCA⋯ 1 CCA � <sup>⋯</sup>

1

1

CCA

n *n*

**113**

*n*

*n*

*n*

*n*

*n*

*n*

*n*

*n*

**9** � �

**8** � �

**7** � �

**6** � �

**5** � �

**1** � �

**1** 1P *i*¼1

**2** 1P *i*¼1

**3** 1P *i*¼1

**4** 1P *i*¼1

**5** 1P *i*¼1

**6** 1P *i*¼1

**7** 1P *i*¼1

**8** 1P *i*¼1

**9** 1P *i*¼1

**Table 1.** *Layout pattern of the* 

*representation*

 *for the first nine binomial coefficients.*

*j*¼1

*i*¼1

*k*¼1

*j*¼1

*i*¼1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*o*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*p*¼1

*o*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*q*¼1

*p*¼1

*o*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

1½ � 8P

*j* P

1 " # 7P

*k*P

1 " # 6P

*j* P

*l* P

*j* P

*k*P

1 " # 5P

P

*m*

*l* P

*j* P

*k*P

1 " # 4P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " # 3P

*o*P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " # 2P

*p*P

*o*P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " # 1P

*q*P

*p*P

*o*P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " #

*j*¼1

*i*¼1

*k*¼1

*j*¼1

*i*¼1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*o*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*p*¼1

*o*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

1½ � 7P

*j* P

1 " # 6P

*k*P

1 " # 5P

*j* P

*l* P

*j* P

*k*P

1 " # 4P

P

*m*

*l* P

*j* P

*k*P

1 " # 3P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " # 2P

*o*P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " # 1P

*p*P

*o*P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " #

*j*¼1

*i*¼1

*k*¼1

*j*¼1

*i*¼1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*o*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

1½ � 6P

*j* P

1 " # 5P

*k*P

1 " # 4P

*j* P

*l* P

*j* P

*k*P

1 " # 3P

P

*m*

*l* P

*j* P

*k*P

1 " # 2P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " # 1P

*o*P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " #

*j*¼1

*i*¼1

*k*¼1

*j*¼1

*i*¼1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*n*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

1½ � 5P

*j* P

1 " # 4P

*k*P

1 " # 3P

*j* P

*l* P

*j* P

*k*P

1 " # 2P

P

*m*

*l* P

*j* P

*k*P

1 " # 1P

*n*P

P

*m*

*l* P

*j* P

*k*P

1 " #

*j*¼1

*i*¼1

*k*¼1

*j*¼1

*i*¼1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

*m*¼

1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

1½ � 4P

*j* P

1 " # 3P

*k*P

1 " # 2P

*j* P

*l* P

*j* P

*k*P

1 " # 1P

P

*m*

*l* P

*j* P

*k*P

1 " #

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation*

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

*j*¼1

*i*¼1

*k*¼1

*j*¼1

*i*¼1

*l*¼1

*k*¼1

*j*¼1

*i*¼1

1½ � 3P

*j* P

1 " # 2P

*k*P

1 " # 1P

*j* P

*l* P

*j* P

*k*P

1 " #

*j*¼1

*i*¼1

*k*¼1

*j*¼1

*i*¼1

1½ � 2P

*j* P

1 " # 1P

*k*P

1 " #

*j* P

*j*¼1

*i*¼1

1½ � 1P

*j* P

1 " #

1½ �

**2** � �

**3** � �

**4** � �

(41)

¼ ð Þ� *k*þ Y 1 1 *f*¼1 X *δ* ∗ *f*�1 *δ* ∗ *<sup>f</sup>* ¼0 X *δ* ∗ *f*�1 1 *δδ* <sup>∗</sup> *f*�1 ¼1 ⋯ X *δ* ∗ *f δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �þ1k*δδ* <sup>∗</sup> *<sup>f</sup>* <sup>þ</sup><sup>1</sup> *δδ* <sup>∗</sup> *f* ¼1 ⋯ X 2 *δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �þ1k*l*1 *k*¼1 ⤸ 8 >>>< >>>: 0 BBB@ 2 6 6 6 4 X 1 *δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �þ1k*k j*¼1 X 0 1 *i*¼1 *ii* " # *j*,*k*,*::*, *δδ* <sup>∗</sup> *f* , … , *δδ* <sup>∗</sup> *f*�1 9 >= >; 1 CA *x δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f f* 3 7 <sup>5</sup>• *<sup>x</sup> δ* ∗ ð Þ� *k*þ1 1 *k*þ1 � � ¼ ð Þ� *k*þ Y 1 1 *f*¼1 X *δ* ∗ *f*�1 *δ* ∗ *<sup>f</sup>* ¼0 *δ* ∗ *f*�1 *δ* ∗ *f* ! ! *x δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f f* 2 4 3 5• *x δ* ∗ ð Þ� *k*þ1 1 *k*þ1 � � <sup>¼</sup> ð Þ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *xk* <sup>þ</sup> *xk*þ<sup>1</sup> *<sup>n</sup>*

(where it was substituted with ð Þ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *xk <sup>n</sup>* according to the hypothesis). This result confirms the hypothesis and the validity for (2).

## **Appendix B: Table**

Next on **Table 1**, the symbolic representations for the binomial coefficients are presented. The value actually comes from the output of the program *coef.mc* provided in Section 3.2, after being evaluated with the corresponding values of *k* and *n*. This table is shown below.

