**4. Conclusions**

With this result an alternative way to represent binomials and multinomies alongside their respective coefficient calculations was exposed. The results obtained from [2] were extended by broading formula (1) to the multinomial instances, by showing that those results can be applied suitably to the theoretical cases of study and by the building up of two algorithms which were implemented in two programs in the CAS Maxima. What will be remaining for a subsequent research work will be 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 or even faster.

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> *xs <sup>n</sup>* <sup>¼</sup> <sup>Y</sup>*<sup>s</sup>*�<sup>1</sup>

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

ð Þ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> *<sup>n</sup>* <sup>¼</sup> <sup>Y</sup><sup>1</sup>

*f*¼1

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

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼0

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>* <sup>¼</sup> <sup>Y</sup>

*δ* ∗ *f*�1 *δ* ∗ *f*

! !

⋯ 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>

X *δ* ∗ *k*�1

8 ><

0

>:

*δδ* <sup>∗</sup> *k*�1 ¼1

1

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

*δ* ∗ 1

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

> *δ* ∗ 2

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

<sup>2</sup> <sup>þ</sup><sup>1</sup>

⋯ X

*δ* ∗

*f*

attempt is to prove that

X *δ* ∗ *f*�1

> X *δ* ∗ 0 1

*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>

*δ* ∗ *<sup>f</sup>* ¼0

8 >< >:

8 >< >:

0 B@

0 B@

2 4

ð Þ� *k*þ Y 1 1

*f*¼1

X*δ* ∗ 0 *δ* ∗ <sup>1</sup> ¼0

X*δ* ∗ 1 *δ* ∗ <sup>2</sup> ¼0

> ⋯ • X *δ* ∗ *k*�1

**111**

*δ* ∗ *<sup>k</sup>* ¼0 *f*¼1

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

X*δ* ∗ *f*�1 *δ* ∗ *<sup>f</sup>* ¼0

X 1

*j*¼1

8 >>><

>>>:

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

0

BBB@

þ1k*k*

X*δ* ∗ *f*�1 *δ* ∗ *<sup>f</sup>* ¼0

X 1

*j*¼1

*n i* � �*xn*�*<sup>i</sup>* <sup>1</sup> *xi* 2

X *δ* ∗ *f*�1 1

8 >>><

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation*

0

BBB@

þ1k*k*

X *δ* ∗ *f*�1 1

*δδ* <sup>∗</sup> *f*�1 ¼1

X 0 1

*i*¼1 *ii* " #

*k*�1

2 4

*f*¼1

3 5• *x δ* ∗ ð Þ� *k*þ1 1 *k*þ1

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

⋯ X

⋯ X 2

> *δ* ∗ *k*

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

*δδ* <sup>∗</sup> *k* ¼1

*k*¼1

*<sup>k</sup>* <sup>þ</sup><sup>1</sup>

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

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

2

X 1

*j*¼1

X 1

*j*¼1

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

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

⋯ ••• ⋯••• ⋯ ••• ⋯••• ⋯

⋯ X

B@ (40)

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

2

*k*¼1

*k*¼1

>>>:

��� <sup>X</sup>

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

> *δ* ∗ *f*

*δδ* <sup>∗</sup> *f* ¼1

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

> 9 >>>=

1

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

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

*f*

>>>;

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

> *δ* ∗ *f*

*δδ* <sup>∗</sup> *f* ¼1

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

> 9 >>>=

1

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

⋯ X

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

2

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

3 5• *x δ* ∗ *k*�1 *k*

� � <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>* (39)

X 0 1

*i*¼1 *ii* " #

X 0 1

*i*¼1 *ii* " #

⤸

*j*,*k*,*::*, *δδ* <sup>∗</sup>

*j*,*k*,*::*, *δδ* <sup>∗</sup>

<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> •⋯

� � (38)

þ1k*l*

*<sup>k</sup>*¼<sup>1</sup> <sup>⤸</sup>

>>>;

⋯ X

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

2

*k*¼1

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

þ1k*l*

(36)

(37)

*δδ* <sup>∗</sup> *f*�1 ¼1

X 0 1

*i*¼1 *ii* " #

��� <sup>X</sup>

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

that actually stands for the binomial expansion for 2�summands; then it is

*δ* ∗ *f*�1 *δ* ∗ *f*

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

! !

correct. Now, fix *s* ¼ *k* on (6) and assume by hypothesis that it is correct.

X *δ* ∗ *f*�1

*δ* ∗ *<sup>f</sup>* ¼0

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