Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Question

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows:

$p_0 = a_0$

$p_1 = a_0x+a_1$

$p_2 = a_0x^2+a_1x+a_2$

$p_n = a_0x^n+a_1x^{n-1}+...+a_n$

...

Where the coefficients are the elements of the sequence. The characteristics of $p_n$ are interesting because $p_n(0)=a_n$ and $p_n(1)=\sum_{n}a_n$

For instance, applied to the Möebius function:

$p_0 = 1$

$p_1 = x-1$

$p_2 = x^2-x-1$

$p_3 = x^3-x^2-x$

$p_4 = x^4-x^3-x^2-1$

...

$p_n(0)=\mu(n)$ ($n^{th}$ element of the Möbius sequence) and $p_n(1)=\sum_{n}\mu(n)$ is the partial summation up to $n$ of the Möbius function, in other words, it is the Merten's function.

This is the graph of the first $100$ polynomials, $p_0$ to $p_{99}$

And this is a zoom of the segment [0,1] so it is possible to see the ramifications of each polynomial from the position of the last $\mu(n)$ to the position of the last $\sum_n\mu(n)$. The shape of the paths is quite curious because it is a representation of the surjective-only diagram of $\mu(n) \to \sum_n \mu(n)$ for each $n$ in $[0,100]$.

I would like to ask the following questions:

1. Is it possible to know something about the properties (e.g. the convergence of the accumulated sum) of a sequence as the $lim_{n \to \infty}p_n(1)=lim_{n \to \infty}\sum_{n}a_n$ by the calculation of the shape of the generic polynomial generated with the elements of the sequence as coefficients? e.g. finding the the shape of the "limit" polynomial $p_n$ when $n \to \infty$?

2. I tried to find some papers about this kind of approach, to understand if it leads to something or it is just visually interesting. Are there any papers regarding the generation of polynomials by using the elements of sequences as coefficients? Thank you!

Answer 1

This really only addresses question 2, since the construction is not exactly what you describe - in particular, the indexing is reversed - but you may be interested in generating functions https://en.wikipedia.org/wiki/Generating_function.

Given a sequence $\mathcal{A}=(a_i)_{i\in\mathbb{N}}$, we can associate to it the formal power series $$a_0x^0+a_1x^1+a_2x^2+...$$ (this is the ordinary generating function). There are many other kinds of generating function we can associate to $\mathcal{A}$ - e.g. the exponential generating function $$\sum_{i=0}^\infty {a_ix^i\over i!}.$$

Often this formal power series will actually be equal (on some open neighborhood) to some function $f$, and by studying $f$ it turns out we can indeed gain information about $\mathcal{A}$. I'm going to stop here, because

• there is a truly gargantuan amount of research about generating functions, and

• I don't know any of it,

but hopefully you find this valuable (and hopefully someone who actually knows things about generating functions stops by to give a better answer!).