README.md

Formula of power sum

This algorithm helps to generate the sum equation of the power function:

Usage

usage: posum.py [-h] [-d DEGREE] [-v] [-s SHOW]

optional arguments:
  -h, --help            show this help message and exit
  -d DEGREE, --degree DEGREE
  -v, --version         show version
  -s SHOW, --show SHOW  only showing the formula (f), triagle matrix (m) or both (b)[default]?

If we want to know the equation of the sum function of the nth power, we run the python script and enter the degree n of the sum function.

python posum.py -d <the degree n>

The result consists of the power sum formula of n degree and the triangle matrix. Besides, wanting to show either. Let's use the argument -s

Example

Need help to use

python posum.py -h

To show version of algorithm.

python posum.py -v

For example, when we want to find the sum function of the power of 6, we run the command and enter the degree of the function as 6.

#input
python posum.py -d 6

#output
Sum formula of x^ 6 is:  +1/42*x^1+0*x^2-1/6*x^3+0*x^4+1/2*x^5+1/2*x^6+1/7*x^7

Matrix triagle is:

         1      2     3     4    5    6    7

  0      1
  1    1/2    1/2
  2    1/6    1/2   1/3
  3      0    1/4   1/2   1/4
  4  -1/30      0   1/3   1/2  1/5
  5      0  -1/12     0  5/12  1/2  1/6
  6   1/42      0  -1/6     0  1/2  1/2  1/7

So the equation sum of the power of the sixth power has the form:

The principle of the algorithm

The algorithm works by forming a triangular matrix with each value of the matrix being a coefficient of the sum equation. The coefficients of the equation of degree n are the values of the n-th row of the matrix. Values of the matrix is calculated according to the system of equations:

$$ \begin{cases} a_{01} = 1 \\ a_{ij} = \dfrac{i}{j} \cdot a_{(i-1)(j-1)}, & \forall i \ge 1,\ \forall j \ge 2 \\ \sum\limits_{k = 1}^{j} a_{ik} = 1, & \forall i, j \end{cases} $$

Application

• Create sum formula of polinomial functions