Trying to understand an abstract definition of power series..

Question

Can anyone explain to me what the $X_{n}$ and $X_{n}^{m}$ are really standing for? Starting from "We write X for the power series..."

I think the book defined, as usual, formal power series as sequences in the ring $R$. Thus $X$ is a sequence and its $n$th term is $X_n$. Now the $n$th term of the power series $X^m$ is $X_n ^m$ — Maxime Ramzi, Apr 17 '17 at 13:44
So is this intended to use an indicator function without mentioning it? I do see the logic here, but i cannot ,yet, articulate the definition. — pigeon, Apr 17 '17 at 14:13
Not really, although in the end $X$ is the indicator function of ${1}$, the idea of the definition is to abstract from problems of convergence in analysis, and from problems of finiteness in polynomials. The sequence $(0,1,0,...)$ is denoted by $X$ because it "represents" the function $x\to x$, and similarly, if $R= \mathbb{R}$ for instance, the sequence $(1,1,1,...)$ is denoted by $\displaystyle\sum_0^\infty X^n$, and it "represents" the function $x\to \displaystyle\sum_0^\infty x^n$ (only defined on a smaller subset). Have you not studied polynomials before ? — Maxime Ramzi, Apr 17 '17 at 14:21
More generally, the sequence $(a_n)$ will be denoted by $\displaystyle\sum_0^\infty a_n X^n$ and looking at the definition, if $m$ is any nonnegative integer, $(\displaystyle\sum_0^\infty a_n X^n)_m= \displaystyle\sum_0^\infty (a_n X^n)_m= \displaystyle\sum_0^\infty a_n X^n_m = a_m$ (this makes sense because the sum is always finite here, indeed it has only one term), which explains the definition of $X^n_m$ (or in your case $X^m_n$) — Maxime Ramzi, Apr 17 '17 at 14:24
@HagenKnaf: well a sequence has an $n$th term doesn't it ? And since $X= (0,1,0,...)$, $X=(\delta_{n,1})n$, so that $X_n = \delta{n,1}= 1$ if $n=1$, $0$ otherwise — Maxime Ramzi, Apr 17 '17 at 14:27
I have studied polynomials many times. I am just trying to follow the definition here, because it is used multiple times and it still unknown to me. Since you can see what the author's intension here, can you give me some guides using that same approache so i can be fully acquainted to it. I am starting to follow better. Most of the texts i find about polynomials define them more readily as polynomials before talking about their product and addition like here. — pigeon, Apr 17 '17 at 14:36
@Max, finally i fully grasped your answer. Thank you very much!!! — pigeon, Apr 17 '17 at 14:40

score 3 · Accepted Answer · edited Apr 17 '17 at 15:00

Just like that of polynomials (not polynomial functions), the definition of formal power series is simply that they're sequences, except unlike polynomials they can be nonzero infinitely many times.

If you see a sequence $(a_n)$ as a power series, it is meant to represent an "infinite polynomial" that would look like this $\displaystyle\sum_{k=0}^\infty a_k X^k$.

The definitions of the product, and the sum of two power series follows from this "interpretation", and so do those of $X$ and $X^n$. Indeed, firstly, assume you want to describe the power series $X$ as a sequence $(a_n)$. Then looking at the infinite sum, you want $a_1=1$, and $a_n = 0$ for $n\neq 1$. Thus you put the $n$th term of $X$ (which is a sequence, so this $n$th term is $X_n$) to be $1$ if $n=1$, $0$ otherwise. Then you do the same for $X^m$, which gives you the formula $X^m_n = 1$ if $n=m$, $ 0$ otherwise (remembering that $X^m_n= (X^m)_n$ is the $n$th term of the sequence $X^m$.

Then one can check that $X^m$ is indeed $X\times...\times X$, $m$ times with the definition of the product that was given.

But first let me explain (I think I understood that you had trouble with this as well) why the product is defined as such. Consider $(p_n)$ and $(q_n)$ two sequences. We want to define their product in such a way that coincides with the interpretation that I gave earlier.

We want $(p_n)(q_n)=(c_n)$ to coincide with $\left(\displaystyle\sum_0^\infty p_nX^n\right)\left(\displaystyle\sum_0^\infty q_nX^n\right) = \displaystyle\sum_0^\infty c_nX^n$.

Let's "calculate" (this is just to explain to you the definition, this is not really rigourous since there is no notion of convergence) the LHS:

$\left(\displaystyle\sum_0^\infty p_nX^n\right)\left(\displaystyle\sum_0^\infty q_nX^n\right)= \displaystyle\sum_{(k,j)\in \mathbb{N}^2}p_k q_j X^{k+j}$, then we rearrange the terms according to the value of $k+j$ which gives $ \displaystyle\sum_{n=0}^\infty \displaystyle\sum_{(k,j)\in \mathbb{N}^2, k+j=n}p_k q_j X^n$. So this gives $c_n = \displaystyle\sum_{k+j=n}p_k q_j$, which can be rearranged to give $c_n =\displaystyle\sum_{k=0}^n p_k q_{n-k}$ .

Next (I'll leave that to you) you can prove, using this definition of product, by induction, that $X$ multiplied $m$ times with itself does give $X^m$ as defined earlier, which justifies the notation. At first, I recommend you keep in mind the fact that you're dealing with sequences, but I think that when you get more experienced you can "forget" that they're sequences, and really "think of them as power series".

Let me know if there's anything I missed

Indeed excellent and just served the purpose of my question. But can you link me to any resource using that approach? — pigeon, Apr 17 '17 at 15:39

Trying to understand an abstract definition of power series..

1 Answers1