# Questions tagged [power-series]

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.

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### Zeta Lerch function. Proof of functional equation.

so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following. In the article "Note sur la function" by Mr. Mathias Lerch, a ...
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### Infinite Series : $1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\ldots$

I need Help to evaluate infinite series : $$S=1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\cdot\cdot\cdot$$ My try: Let $$c_n:= \left({\frac{1-i}{2}}\right)(i)^n+\left({\frac{1+i}{2}}\right)(-i)^n$$ ...
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### A problem of determining whether a power series belongs to $\mathbb{C}(u)$

I am reading a paper "Drinfeld coproduct, quantum fusion tensor category and applications" and I have a probelm. Here is the arxiv:Drinfeld coproduct, quantum fusion tensor category and ...
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### Solving $y''(x)-2xy'(x)+y(x) = 0$ using power series

I'm solving $y''(x) -2xy'(x) +y(x) = 0$ using the power series ansatz $y(x) = \sum_{n=0}^{\infty}a_n x^n$ . Plugging in I get: \bigg(\sum_{n=0}^{\infty}n(n-1)a_nx^{n-2}\bigg)-2x\bigg(\...
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### Definition and Use of the Schett Polynomial in the Jacobi Taylor Series

I am having a tough time understanding the definition and use of the Schett polynomial introduced in the paper here. I have two questions related to this polynomial. My first question concerns its ...
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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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