0

I am somewhat confused between the difference of power series, formal power series and convergent power series. Where is the difference between these three definitions born? And in particular, where exactly are the generating functions?

Luis Alexandher
  • 1,260
  • The first two are the same, where the added "formal" is only to emphasize that we care about the algebraic object, which you can think of as sequences of numbers with certain addition and multiplication on them. The $x^n$ is only indicating the index of the element of the sequence and the $x$ is not intended to be replaced by a value. Perhaps the only evaluation is putting $x=0$, which corresponds to the operation of taking the coefficient of $x^0$. One can define an operation that corresponds to replacing $x$ with another power series that has the coefficient of $x^0$ equal to $0$. – plop Sep 19 '22 at 19:21
  • 1
    There is a topology in which you can think of the formal sum as the limit of the partial sum. See here, but you can ignore this, if you are not familiar the notions involved there. – plop Sep 19 '22 at 19:22
  • Convergent power series are series in which one is interested in studying values obtained when replacing $x$ with a non-zero value. The coefficients will need to have some notion of convergence, to give meaning to the "infinite sum" as a value in the set of coefficients. – plop Sep 19 '22 at 19:24
  • ok, then, since we are not interested in studying the generating functions at all, the convergence, if not the coefficients, can I say that these are formal power series? – Luis Alexandher Sep 19 '22 at 19:29
  • What do you mean by not interested in studying the generating functions? The algebraic properties of generating functions are the algebraic properties of their formal power series, and those give information about the sequence of coefficients. – plop Sep 19 '22 at 19:31
  • sorry, I made a mistake, I mean that in the case of generating functions we are not interested in studying convergence – Luis Alexandher Sep 19 '22 at 19:35
  • I took a semester long course on Analytic Combinatorics, a few details are in my answer, far more in the book/wiki – Alan Sep 19 '22 at 19:37
  • I don't know. It depends. If the power series in question do converge for some non-zero values of $x$ the study of their values (for example using the theory of analytic function) can produce non-trivial information about the sequence of coefficients, like asymptotics. – plop Sep 19 '22 at 19:37

1 Answers1

1

A power series is just an infinite sum of powers of $x$ with some coefficients.

$$\sum_{i=o}^\infty a_ix^i $$ is an example for a single variable version centered at 0.

A generating function is one that gives us the terms $a_i$ as one would in a Taylor series, i.e. if the generating function is $g(x)$, that makes $a_i=\frac {g^{(i)}(0)}{i!}$

Normally when we use these as functions, it only makes sense to talk about them in areas where the infinite sum converges, so it is a convergent power series for any value of $x$ that makes the sequence converge, usually in some radius of the center. (Radius of convergence) When it converges, we can then talk about it as a function, $$f(x)=\sum_{i=o}^\infty a_ix^i $$ $$$$ A formal power series is one in which we don't care about looking it as a function, so we don't care if it converges or not. In the area of analytic combinatorics, we can use generating functions to find the answers to various combinatorics problems as the coefficients of formal power series. https://en.wikipedia.org/wiki/Analytic_Combinatorics is the wiki on a really crazy textbook on the subject. It turns out we can use complex analysis to develop tight asymptotic bounds on discrete counting problems

Alan
  • 16,582
  • Precisely my doubt stems from Flajolet's book, in this case (from analytical combinatorics) I understand that when the generating function does not "converge" then asymptotic analysis is used, or am I wrong? – Luis Alexandher Sep 19 '22 at 19:40
  • 1
    @LuisAlexandher It's more that we just don't care if it converges or not, we're not viewing them as functions at all. We're using them because the coefficients give us our combinatorics and we can do the asymptotic analysis on it, whether it converges or not – Alan Sep 19 '22 at 19:44
  • Ok, so I can't classify the generating functions as formal power series or as convergent power series? – Luis Alexandher Sep 19 '22 at 19:51
  • or why the term generating function has the word "function"? – Luis Alexandher Sep 19 '22 at 19:56
  • 1
    The generating function is the one that gives us the sequence $a_n$ by taking repeated derivatives. That's a function, the resulting power series we don't care if it defines a function or not – Alan Sep 19 '22 at 20:06