Methods

Integration

I wrote a python script to calculate the sequences with the df(x) = f(x) - (x - 1) definition of a differential.

#!/bin/python

size = 11
power = 9

# initialize the sequence with 0
seq = {}
for i in range(-size, size):
    seq[i] = 0

# the first term in a sequence, 0 by default
c = {}
for p in range(0, power + 1):
    c[p] = 0
# specific terms
c[0] = 1

# # x^4
# power = 4
# c[0] = 24
# c[1] = -36
# c[2] = 14
# c[3] = -1
# c[4] = 0

for p in range(0, power + 1):
    term = c[p]
    print(term)
    for i in range(1, size):
        term += seq[i]
        seq[i] = term
        print(seq[i])
    print("endl")

I run it with the following one-liner to produce nice tables:

$ ./script.py | tr '\n' ' ' | tr "endl" "\n" | column -t

1  1  1   1   1    1    1     1     1      1      1
0  1  2   3   4    5    6     7     8      9      10
0  1  3   6   10   15   21    28    36     45     55
0  1  4   10  20   35   56    84    120    165    220
0  1  5   15  35   70   126   210   330    495    715
0  1  6   21  56   126  252   462   792    1287   2002
0  1  7   28  84   210  462   924   1716   3003   5005
0  1  8   36  120  330  792   1716  3432   6435   11440
0  1  9   45  165  495  1287  3003  6435   12870  24310
0  1  10  55  220  715  2002  5005  11440  24310  48620

To derive the expressions of the sequences, I solve simultaneous equations with the terms after the initial one. The power variable corresponds to the degree of the polynomial. The above code generates Pascal's triangle. To find the third integral/sequence, you would need to solve for a polynomial of the second degree.

s(x) = ax^2 + bx + c

The numbers in the table represent s(x) at a specific x (the place of the term in the sequence).

We know c

s(0) = c = first term of the sequence

and want to find a and b. From the table we get these simultaneous equations:

s(1) = a(1)^2 + b(1) + c = 1
s(2) = a(2)^2 + b(2) + c = 4

so

a(1)^2 + b(1) = 1 - c
a(2)^2 + b(2) = 4 - c

c = 0 in this case

a(1)^2 + b(1) = 1
a(2)^2 + b(2) = 4
2a + b = 1
4a + 2b = 4
a = 1/2
b = 1/2

And the final expression becomes:

s(x) = (1/2)x^2 + (1/2)x

Factorized:

s(x) = (1/2)(x + 1)(x + 0)

Differentiation

The program I wrote to find the differentials of the x^n sequences is, sadly, not so nice and simple so I won't post the code here. You can get it by running:

$ git clone git://git.yotsev.xyz/differentials.git

You get similar tables from it to the one above and follow the same algorithm to find the expressions of a given differential. The only difference is that the initial term is not the leftmost one but the one bellow the 0 of the initial sequence.

You can also switch between the two definitions of a differential by supplying an argument to the program:

$ ./prog prior

for the one above and

$ ./prog post

for the other one. It uses the post one by default.