Recently I've been trying to make sense of generating functions by trying to create an interpretation for it, so far I've made interpretation of addition, multiplication, division and derivative and integrals. However, there are some parts which I can't explain intuitively. Below, I explain the language I have derived for it already.
0. Meaning of monomials and polynomials
The meaning of monomials, is a number of items. For example, $x^n$ means a list of 'n' items. A polynomial is a collection of a number of different items.
1. Addition
Addition is the way of saying 'or' using polynomials. So, if we had $ x^p +x^q$ that would mean a list of 'p' items or a list of 'q' items
2. Multiplication
Multiplication of polynomials, means to do choose 'decisions' consecutively. Take for example $ (x^3 + x) (x)$ , in the first factor, I have a choice of three items $ (x^3)$ or one items $(x)$ and when I multiply either these onto the one item (x) in the second factor, the resultant polynomial is the collection of ways to choose 'n' items by choosing from the set in the first fact and then the one in the second.
For instance, multiplying the $x^3$ in the first factor with the $ x$ of the second is equivalent to choosing three items then choosing one item, hence four in total
3. Division
Division of polynomials is interpreted as removal of items, for example, $$ \frac{x^2 +x}{x} $$ represents removing one element from a set of two or one item. We have, $x+1$ as result.
4.Derivatives
Derivatives represent removal of items, for example if I have a polynomial $x^3$ it would represent a list of three items. Maybe like {apple,banana , orange} , it's derivative $3x^2$ represents that you get three lists of two items if you randomly deleted a random item from the previous set.
5. Integrals
Opposite to derivatives, integrating a polynomial is equivalent to adding back in items to a set.
Remark:
We can write the idea of derivative through the idea of removal, we have:
$$ Dx^n = n x^{n-1} = n \frac{x^n}{x}$$
We have, remove one item from the list and duplicate $n$ times.
Problems with my interpretation:
- I can't reason negative coefficient of polynomial in this combinatoric view point of polynomials( ex: what does -3x mean in this new language I said before)
- What would decimal coefficents mean? like (2.1 x) ?ex: exponential generating functions
- I have yet to find a 'good enough' interpretation of remainders we get from dividing polynomials
- For taking derivative thing, I considered my $x^3$ is a collection of three distinct items, but $x^n$ as itself doesn't have a sense of distinctness other than the size of the set denoted by the exponent
References:
- https://www.youtube.com/watch?v=BZA5_D789Fk&list=PLKc2XOQp0dMwj9zAXD5LlWpriIXIrGaNb&index=16 (around 6:00 of this vid)
Maybe helpful: http://strictlypositive.org/diff.pdf
