# Questions tagged [formal-power-series]

This tag is for questions relating to "formal power series" which can be considered either as an extension of the polynomial to a possibly infinite number of terms or as a power series in which the variable is not assigned any value.

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### Connection Between Derivations of Finite and Infinite Binomial Expansion

At first when learning the binomial expansion you learn it in the case of working as a shortcut to multiplying out brackets - anti-factorising if you will. In these cases what you are expanding takes ...
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### Seeking a Purely Formal Power Series Solution

Seeking a Purely Formal Power Series Solution When I read about generating functions, I encountered the following problem: Suppose that the set of nonnegative integers is partitioned into a finite ...
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### Can Zeckendorf's theorem be proven using generating functions?

First, I state Zeckendorf's theorem. Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum ...
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### Compatibility - Topological modules contra vector spaces

So Tréves in his book on topological vector spaces shows that a filter $\mathcal{F}$ on a $\textit{vector space}$ $E$ is the filter of neighbourhoods of zero compatible with the linear structure if ...
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1 vote
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### Product Principle for Generating Functions?

Product Principle for Generating Functions I am inquiring about the Product Principle for Generating Functions as applied in combinatorial counting problems. First, let me state the principle: ...
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### Counting irreducible Permutations

Let $x:=(x_1,...,x_n)$ be a permutation of $\{1,...,n\}$. We say, $x$ is irreducible iff $\{x_1,...,x_m\}\neq\{1,...,m\}$ for $1\leq m \leq n-1$. Let $g(n)$ be the number of irreduible permutations of ...
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### If $f(g(x))=x$, does $g(f(x))=x$ hold?

If $f(x)$ and $g(x)$ are formal power series, i.e. $$f(x)=\sum_{n\ge 0} a_n x^n, g(x)=\sum_{n\ge 0} b_n x^n,$$ and $$f(g(x))=x,$$ can it be proved that always have $$g(f(x))=x?$$ It seems intuitive, ...
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### A generalization of integration by parts

Many years ago, I came up with a short generalization of integration by parts that was definitely known, but I could never find a reference for it. I was considering throwing it on arxiv, but before I ...
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### Building the theoretical foundation for generating functions - formal power series

I have read several documents on generating functions. I would like to inquire about two issues: Among the materials I have read, some mention generating functions constructed from formal power ...
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