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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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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Fubini for formal power series
Let $A_{k,n} \in \mathbb{C}[[X]]$ such that $\lim _{k+n \rightarrow \infty} A_{k, n}=\mathbf{0}$. Prove that
$$
\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} A_{k, n}=\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} ...
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Formal power $ \sum_{k=1}^{\infty} A_k = \sum_{k=1}^{\infty} A_{\pi(k)}. $
Consider the sequence of elements $A_1, A_2, \cdots \in \mathbb{C}[[X]]$ satisfying $\lim A_n = \mathbf{0}$ and $\pi: \mathbb{N} \rightarrow \mathbb{N}$ being a bijection. Then,
$$
\sum_{k=1}^{\infty} ...
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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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$\forall a,b \in K[[x]]: a | b \implies v(a) \leq v(b)$ where $v$ is the valuation
The valuation of a power series is the smallest index $i$ of the coefficients $a_i$ so that $a_i \ne 0$.
Since $a|b$ there exists a $c\in K[[x]]$ so that
$$\sum_{n=0}^\infty a_nx^n\sum_{n=0}^\infty ...
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find square root of $x^2+x^3 $ in formal power series $k[[x,y]]$
I am trying to show that the polynomial $y-x^2-x^3$ is reducible in the formal power series ring $k[[x,y]]$. I am attempting the question by finding a polynomial in $k[[x,y]]$ which is the square root ...
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Compute the power series of $\frac{2-6x+x^2}{1-3x}$
I'm struggling with the following:
Suppose that we have a formal power series $\sum_{n \geq 0} a_nx^n$ which is equal to the fraction $\frac{2-6x+x^2}{1-3x}.$ I want to find an explicit formula for $...
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Convergence of $f(x)= \sum_{n=0}^{\infty} x^{2^n}$ [duplicate]
If we define $f(x)= \sum_{m=0}^{\infty} x^{2^m} = \lim_{n\to\infty}f_n(x)= \lim_{n\to\infty}\sum_{m=0}^{n} x^{2^m}$.
It's easy to see that $|{f(x)}|$ converge in the interval $(-1,1)$. (We can easily ...
