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I have read several documents on generating functions. I would like to inquire about two issues:

  1. Among the materials I have read, some mention generating functions constructed from formal power series (an algebraic structure). In this approach, operations are developed similar to those for power series in analysis. Then, when necessary, they view generating functions as a power series (in the analytical sense) over a domain where the series converges. A few materials begin with defining generating functions as power series in analysis, over a domain where the series converges. I want to ask which approach truly constitutes the starting point for generating functions. Furthermore, why are there these two approaches? What significance do they hold? I still do not understand.

  2. Dirichlet generating functions are also considered a type of generating function, yet they are not any form of formal power series. So, ultimately, what is a generating function and how many types are there?

Somos
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Math_fun2006
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    For number 1.: You usually define the generating function as formal power series, if it converges you you might be able to use analysis to simplify expressions. For number 2.: Same story, you usually define formal Dirichlet generating functions and if they converge you might be able to use complex analysis to say something you want to say. Those are the two most common types of generating function. –  May 08 '24 at 16:07
  • For what are they I think best is to look at small examples, look here for example https://math.stackexchange.com/questions/4911500/4-die-sum-equals-20-how-to-generalise-it-to-n-die#comment10486332_4911500 –  May 08 '24 at 16:09
  • I like the way of constructing from from formal power series, it provides a solid foundation for generating functions, compared to the second method. However, with that construction, I don't have a way to build Dirichlet generating functions.

    My construction method is similar to the one described in the article here https://link.springer.com/article/10.1365/s13291-022-00256-6

     – Math_fun2006 May 08 '24 at 16:12
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    You can see formal Dirichlet series at wikipedia https://en.wikipedia.org/wiki/Dirichlet_series#Formal_Dirichlet_series . The most famous example of the use of Dirichlet generating functions is that of prime numbers. You look at the Dirichlet generating function of the von Mangoldt function (which converges in some region) and use complex analysis to get asymptotics for it and then from that you get asymptotics for the prime numbers and get the prime number theorem. –  May 08 '24 at 16:18
  • Thank you, of course, I have seen and understood them. I am asking about how to build the theoretical foundation if starting from formal power series for Dirichlet generating functions. If starting from analysis, I have nothing to say (I reiterate, ordinary generating functions and exponential generating functions are built from formal power series, so I want to ask if that can be done for Dirichlet generating functions). – Math_fun2006 May 08 '24 at 16:29
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    I linked formal Dirichlet series and gave an example on where they are used. I don't understand what you are asking. –  May 08 '24 at 16:31

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