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December 9th, 2017, 10:24 PM   #1
jks
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Math Focus: Electrical Engineering Applications
Method of finding polynomial series coefficients?

Recently, at work, an application came up where I had a series that I suspected was generated by successive terms of a polynomial and I needed to find the coefficients of that polynomial, similar to the problem in this thread (Reference 1). I give another sample calculation in Reference 1, with a little bit more explanation than presented below relating to the coefficients of the numbers raised to a power.

I was aware of taking the differences until a constant is reached (due to previous posts on this forum), which gives the order of the polynomial. I did this and used the linear equations to get the result (there was a cubic term and a linear term).

Then I started playing around and I think I found a way to get the coefficients from the difference triangle. The method relies on two properties of Pascal's Triangle (PT), which I ask about here (Reference 2).

While I am not certain of these properties of PT for all rows, I have calculated them to be true up to row 100 (for the values that I am interested in) for the purposes of the method that I am about to describe. This is a lot higher than any practical polynomial degree that I would need to solve.

Anyway, I would like to attempt to describe the method and show by a fourth order polynomial why it just might work.

---------------------------------

So, suppose we have the following series given by row 0, and the differences:

$\displaystyle \large {\begin{array}{c c}
& n & 1 & & 2 & & 3 & & 4 & & 5 & & 6 \\
row \\
0 & & 28 & & 101 & & 320 & & 823 & & 1796 & & 3473 \\
1 & & & 73 & & 219 & & 503 & & 973 & & 1677 \\
2 & & & & 146 & & 284 & & 470 & & 704 \\
3 & & & & & 138 & & 186 & & 234 \\
4 & & & & & & 48 & & 48 \\
\end{array}}$

We can see that we have a fourth order polynomial since 4 subtractions were needed to get a constant term (I would take more terms just to be sure in a series that I was not familiar with). So we are looking for:

$\displaystyle \large a \cdot n^4 + b \cdot n^3 + c \cdot n^2 + d \cdot n +
e$

I am going to go term by term to get to the first (left-most) constant of 48, and honestly, you can probably skip on down to where you start seeing braces inside $\LaTeX$.

----------

$\displaystyle \large {\begin{align*} 973 &= a \cdot 5^4 + b \cdot 5^3 + c \cdot 5^2 +d \cdot 5 +e - \left (a \cdot 4^4 + b \cdot 4^3 + c \cdot 4^2 +d \cdot 4 +e \right) \\
&= a (5^4 - 4^4) + b (5^3-4^3)+c(5^2-4^2)+d
\end{align*}}$

----------

$\displaystyle \large 503 = a (4^4 - 3^4) + b (4^3-3^3)+c(4^2-3^2)+d$

----------

$\displaystyle \large 470 = a (5^4 - 2 \cdot 4^4 + 3^4) + b (5^3 - 2 \cdot 4^3 + 3^3)+c(5^2 - 2 \cdot 4^2 + 3^2)$

----------

$\displaystyle \large 219 = a (3^4 - 2^4) + b (3^3-2^3)+c(3^2-2^2)+d$

----------

$\displaystyle \large 284 = a (4^4 - 2 \cdot 3^4 + 2^4) + b (4^3 - 2 \cdot 3^3 + 2^3)+c(4^2 - 2 \cdot 3^2 + 2^2)$

----------

$\displaystyle \large 186 = a (5^4 - 3 \cdot 4^4 + 3 \cdot 3^4 - 2^4) + b (5^3 - 3 \cdot 4^3 + 3 \cdot 3^3 - 2^3)+c(5^2 - 3 \cdot 4^2 + 3 \cdot 3^2 - 2^2)$

----------

$\displaystyle \large 73 = a (2^4 - 1^4) + b (2^3-1^3)+c(2^2-1^2)+d$

----------

$\displaystyle \large 146 = a (3^4 - 2 \cdot 2^4 + 1^4) + b (3^3 - 2 \cdot 2^3 + 1^3)+c(3^2 - 2 \cdot 2^2 + 1^2)$

----------

$\displaystyle \large 138 = a (4^4 - 3 \cdot 3^4 + 3 \cdot 2^4 - 1^4) + b (4^3 - 3 \cdot 3^3 + 3 \cdot 2^3 - 1^3)+c(4^2 - 3 \cdot 3^2 + 3 \cdot 2^2 - 1^2)$

--------------------

$\displaystyle \large { \begin{align*}48 &= a \ \underbrace{(5^4 - 4 \cdot 4^4 + 6 \cdot 3^4 - 4 \cdot 2^4 + 1^4)}_{4! \text{ per Reference 2}} \\
&+ b \ \underbrace{(5^3 - 4 \cdot 4^3 + 6 \cdot 3^3 - 4 \cdot 2^3 + 1^3)}_{0 \text{ per Reference 2}} \\
&+c \ \underbrace{(5^2 - 4 \cdot 4^2 + 6 \cdot 3^2 - 4 \cdot 2^2 + 1^2)}_{0 \text{ per Reference 2}} \end{align*}}$

So:

$\displaystyle a=\frac{48}{4!}=2$

--------------------

Back to 138 with $a=2$:

$\displaystyle \large { \begin{align*} 138 &= \underbrace{2(4^4 - 3 \cdot 3^4 + 3 \cdot 2^4 - 1^4)}_{120} \\
&+ b \ \underbrace{(4^3 - 3 \cdot 3^3 + 3 \cdot 2^3 - 1^3)}_{3! \text{ per Reference 2}} \\
&+ c \ \underbrace{(4^2 - 3 \cdot 3^2 + 3 \cdot 2^2 - 1^2)}_{0 \text{ per Reference 2}}
\end{align*}}$

So to get $b$:

$\displaystyle \large b=\frac{138-120}{3!}=3$

--------------------

Back to 146 with $a=2$ and $b=3$:

$\displaystyle \large { \begin{align*} 146 &= \underbrace{2(3^4 - 2 \cdot 2^4 + 1^4) + 3(3^3 - 2 \cdot 2^3 + 1^3)}_{136} \\
&+ c \ \underbrace{(3^2 - 2 \cdot 2^2 + 1^2)}_{2! \text{ per Reference 2}}
\end{align*}}$

So to get $c$:

$\displaystyle \large c=\frac{146-136}{2!}=5$

--------------------

$\displaystyle \large 73 = 2(2^4 - 1^4) + 3(2^3-1^3)+5(2^2-1^2)+d$

$\displaystyle \large d=7$

--------------------

$\displaystyle \large 28 = 2+3+5+7+e$

$\displaystyle \large e=11$

---------------------------------

So the generating polynomial is:

$\displaystyle 2 \cdot n^4 + 3 \cdot n^3 + 5 \cdot n^2 + 7 \cdot n + 11$

which is easily verified.
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